Thin-sheet creation and threshold pressures in drop splashing

TL;DR: In this paper, high-speed imaging was used to identify a threshold velocity that limits the times at which a thin sheet can be created, which determines the threshold pressure below which splashing is suppressed.

Abstract: A liquid drop impacting a smooth solid substrate splashes by emitting a thin liquid sheet from near the contact line of the spreading liquid. This sheet is lifted from the substrate and ultimately breaks apart. Surprisingly, the splash is caused by the ambient gas, whose properties dictate when and if the sheet is created. Here, I focus on two aspects of this process. Using high-speed imaging I find that the time of thin-sheet creation displays a different quantitative dependence on air pressure if the sheet is created during the early stages of spreading, rather than when the liquid has already spread to a large radius. This result sheds light on previously observed impact velocity regimes. Additionally, by measuring impacts of drops on surfaces comprised of both rough and smooth regions, I identify a new threshold velocity that limits the times at which the thin sheet can be created. This velocity determines the threshold pressure below which splashing is suppressed.

Summary

INTRODUCTION

• Experiments have found that drops splash only above a certain gas pressure in a variety of systems [1, 2, 3, 4, 5], and have provided insight into the liquid and air dynamics during splashing [6, 7, 8, 9].
• When the liquid finally makes contact with the substrate [7], the air directly beneath the drop is trapped in a small bubble [15] and does not further influence the splashing process [16].
• Next, the liquid spreads radially outward in the form of a liquid sheet that remains in direct contact with the substrate, as shown in Fig. 1.1a [6, 16, 17].
• The majority of present theories of splashing do not take this mechanism into account.
• Together, the air-dependent time of thin-sheet creation and the air-independent threshold velocity, below which thin-sheet creation is suppressed, form a pair of necessary and sufficient conditions for thin-sheet creation.

EXPERIMENTAL DETAILS

• The experiments were conducted with a variety of liquids.
• Ethanol and silicone oils (PDMS, Clearco Products) were used to vary the drop viscosity from µ = 1.2 to 48mPa s, while keeping the surface tension approximately constant between σ = 18.7 and 21.6mN m−1.
• The time of etching was chosen to produce a root-mean-square roughness Rrms = 1.9µm, sufficient to prevent thin-sheet creation for the liquids used [5].
• This accuracy was provided by ultra-fast interference imaging, which measures the interference between the light reflected from the bottom surface of the spreading liquid and the top surface of the substrate [16].
• The thinnest air gap the authors can reliably detect, approximately 30nm, is set by the wavelength of the light source (ThorLabs LED, λ = 625nm), the sensitivity of the camera, and the exposure time.

TIME OF THIN-SHEET CREATION

• The thin-sheet creation time depends on a number of parameters, in particular on the ambient gas pressure [6].
• The phase diagram can be understood by considering distinct thresholds for small-r∗ and large-r∗ sheets.
• In the low velocity regime, the small-r∗ sheet persists to lower pressures and Psheet = Psmall-r∗ and the dependence of sheet threshold pressure on impact velocity follows from the properties of small-r∗ sheet creation.
• There, a thin sheet begins to be created, despite the fact that t > tsheet, while the part of the drop that remains in the rough region continues to spread smoothly on the surface.
• The ustop pressure independence has an important practical consequence, as it allows one to measure ustop by simply measuring usheet on a smooth surface.

DISCUSSION AND CONCLUSIONS

• The cause of the sharp transition between small-r∗ and large-r∗ sheets can potentially be understood by considering the geometry of the drop.
• Notably, ucritical and ustop differ with respect to surface tension.
• Further research in both forced wetting and splashing is necessary before the role of surface tension in both processes can be understood.
• Existing experiments [7] and a recent simulation [10] confirm that early during drop impact the contact line does in fact behave differently in the early and late stages of spreading.
• Here, I showed that the threshold pressure is set by two distinct conditions: the air-dependent time of thin-sheet creation, and the air-independent threshold velocity ustop that is related to contact line stability.

Figures

Figure 4.3: A log-log plot of the threshold velocity ustop vs. liquid viscosity µL for D = 3.3mm drops impacting a glass slide at 3.4m s in an atmosphere of air. Various liquids were used: ethanol ( ), silicone oil ( ), a solution of ethanol, water and sucrose ( ), and a solution of water and glycerol ( ).

Figure 3.2: Interference images of 9.4mPa s silicone oil drops of radiusR = 1.65mm impacting a glass slide with velocity of 2.2m s−1 as seen from below. The images were taken at t = 0.08, 0.18, 0.24, and 0.46ms after impact and at three different pressures, which were chosen to illustrate the transition in Fig. 3.1: P = 55kPa below the transition pressure, 58kPa in the transition region, and 70kPa above the transition pressure. At the earliest time the image of the spreading liquid is black, indicating that it is in contact with the glass. A trapped air bubble can be seen trapped at the center of each impact [15]. At t = 0.18ms an interference pattern is seen at the edge of the 70kPa drop, indicating that the liquid is now spreading over a thin air gap and that a thin sheet has been created. The air gap is also seen in parts of the liquid edge at 58kPa. At t = 0.24ms the interference patterns grow for both the 58kPa and the 70kPa drops. In the 58kPa images, the regions of the spreading drop that had not already formed a thin sheet remain on the substrate and do not form a thin sheet until a later time. At t = 0.46ms an air gap has finally developed for the 55kPa drop, as well as the remaining regions of the 58kPa case. Close inspection reveals that the liquid locally bridges the air gap near the contact line, as described in [6]. The white scale bar is equivalent to 3.3mm, the diameter of the original drop.

Figure 4.1: The velocity of the spreading liquid at the splashing onset usheet (open symbols) and the threshold velocity ustop (closed symbols) vs. ambient pressure for R = 3.2mm silicone oil drops of viscosity 9.4mPa s ( ), 19mPa s ( ), and 48mPa s ( ) impacting a glass slide at 3.4m s−1. While usheet decreases with pressure, ustop remains approximately constant. The threshold pressures Psheet, marked by dashed lines of the respective color, are set by the crossover of usheet (P ) and ustop.

Figure 1.1: Successive images of a 9.4mPa s silicone oil drop of diameter D = 3.3mm impacting a glass slide at V = 3.4m s−1 (Re = 1100, We = 1600) at atmospheric pressure. Images (a)-(d) show the drop splashing on a smooth glass slide at times t = 0.18, 0.33, 1.2 and 2.3ms. The red arrow in (b) points to the newly-created thin sheet that grows in subsequent images. Images (e)-(f) show the corresponding frames of a drop impacting a slide that is comprised of a rough (right, dark) and a smooth (left, bright) region. Thin-sheet formation and splashing take place only in the smooth region.

Figure 3.1: Sheet creation time vs. pressure for 9.4mPa s silicone oil drops of radius R = 1.65m s−1 impacting a glass slide with velocity 1.4m s−1 ( ), 2.2m s−1 ( ), and 3.4m s−1 ( ). The increase in tsheet with decreasing pressure is smooth for both the fastest and the slowest drops. In contrast, a transition in tsheet is seen at P = 58 ± 1kPa for the intermediate velocity drop. Above the transition the sheet creation time slowly increases from 0.06ms to 0.12ms as pressure is reduced from 100kPa to 60kPa. As the pressure is further reduced to 57kPa, tsheet more than doubles to 0.27ms. The distribution of tsheet is bimodal in this region, with sheet creation times clustered around either 0.12ms or 0.27ms. As pressure is reduced further, tsheet increases smoothly. The inset shows the transition region in detail. Lines are guides to the eye.

Figure 3.4: Diagram of drop impact outcomes for 9.4mPa s silicone oil drops of radius R = 1.65mm as gas pressure and impact velocity are varied: small-r∗ sheet (red), large-r∗ sheet (blue), and no sheet (white). (Data was not taken below V = 1m s−1, therefore that region is left blank.) The boundaries between the regions are Psmall-r∗ ( ) and Plarge-r∗ ( ). The smaller of the thresholds at a given V determines Psheet (filled symbols). If Psmall-r∗ > Plarge-r∗ , Psmall-r∗ marks the transition between a small-r ∗ and a large-r∗ sheet (empty symbols), as shown in Fig. 3.1. The lines serve as a guide to the eye.

Figure 3.5: Threshold pressures of small-r∗ and large-r∗ sheets for silicone oil drops of viscosity 9.4mPa s and radius R = 0.85mm ( ), 1.20mm ( ), 1.65mm ( ), and 4.6mPa s drops of radius 1.65mm ( ). The open symbols represent the transition between the smallr∗ and large-r∗ sheets. The data is scaled by R0.5.

Figure 4.4: Threshold velocity ustop vs. impact velocity for for 9.4mPa s silicone oil drops of radius 1.6mm ( ). A discontinuous transition separates a regime where ustop is dependent on a low impact velocity and one where ustop is independent of high impact velocities. The two regimes correspond to the small-r∗ and large-r∗ sheets. The dashed lines are a guide to the eye. Inset: ustop vs. V scaled by

Figure 4.2: The velocity of the advancing liquid at the splashing onset usheet vs. gas pressure for D = 3.3mm silicone oil drops of viscosity 4.6mPa s impacting a smooth glass slide at 3.5m s−1 in an atmosphere of air, neon, and helium. The dashed line marks ustop = 2.6m s−1. For each gas, the leftmost points represent the measured Psheet.

Figure 3.3: The radius of region wetted by the drop at tsheet rescaled by drop radius, r ∗ vs. pressure for 9.4mPa s silicone oil drops of radius R = 1.65mm impacting a glass slide with velocity: 1.4m s−1 ( ), 1.9m s−1 ( ), 2.2m s−1 ( ), 2.5m s−1 ( ), and 3.4m s−1 ( ). The transition occurs at r∗ ≈ 1. Lines are guides to the eye. The images show the difference in the shape of a V = 1.8m s−1 drop at times between 0.1ms and 1.5ms, as seen from the side.

Full text

THE UNIVERSITY OF CHICAGO
THIN-SHEET CREATION AND THRESHOLD PRESSURES IN DROP SPLASHING
A DISSERTATION SUBMITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
DEPARTMENT OF PHYSICS
BY
ANDRZEJ LATKA
CHICAGO, ILLINOIS
DECEMBER 2016

Copyright
c
 2016 by Andrzej Latka
All Rights Reserved

To my mother, Maria Latka: I owe nobody more.
To my father, Miroslaw Latka: my role model for physics and life.
To my sister, Agnieszka Latka: my first and closest friend.
To my grandmother, Maria Latka: the strongest person I have ever known.
To my grandfather, Henryk Latka: for showing me how to be a man.
To my grandmother, Anna Lorenz: for the memories carried by the smell of the best apple
pie in the world.
To my grandfather, Roman Lorenz: for teaching me the importance of history.
To Nicholas Minutillo and his family: for becoming my American family.
To Michal Niewiara, a brother.
To Ryszard Chytrowski: for everything.
With all my love to Agnieszka Wergieluk: for making me a better physicist than I would
have been and for making me a happier person than I ever thought I could be.

TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 EXPERIMENTAL DETAILS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 TIME OF THIN-SHEET CREATION . . . . . . . . . . . . . . . . . . . . . . . . 6
4 THRESHOLD VELOCITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . 22
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
iv

LIST OF FIGURES
1.1 Images of a splashing drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1 Time of thin-sheet creation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Interference imaging of sheet creation . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Drop shape at time of thin-sheet creation . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Diagram of drop impact outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Threshold pressures of small-r
∗
and large-r
∗
sheets . . . . . . . . . . . . . . . . 12
4.1 Plot of u
sheet
and u
stop
vs. pressure . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Effect of changing the ambient gas . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Threshold velocity u
stop
vs. liquid viscosity . . . . . . . . . . . . . . . . . . . . 19
4.4 Threshold velocity u
stop
vs. impact velocity . . . . . . . . . . . . . . . . . . . . 20
v