1. Introduction

The contact angle hysteresis is a fundamental hydrophysical phenomenon which appears at the contact of liquid drops and a solid surface [1,2]. It reveals itself in the inequality of the advancing contact angle of θ_A and the receding contact angle θ_R of the drop on this surface.

Hysteresis of contact angles can be recorded in different configurations [3,4]. The most technically simple configuration is the configuration in which the study of static and dynamic behavior of the drop is carried out on the inclined plane [5 - 7]. The study of hysteresis on the inclined plane can be carried out at the constant drop volume. Then, it is always true θ_A > θ_R in the statics conditions for contact angles both in the case of wetting and non-wetting. However, sometimes the inversion of this inequality is possible at the drop sliding up on the inclined plane [8].

Recently, a new direction in the study of wettability and capillary phenomena has been developed. It is elastocapillarity in which the liquid behavior at the contact with elastic deformed surfaces is studied [9 - 11]. In the frames of this direction, the behavior of liquid sessile drops on the plane horizontal stretchable rubber surfaces was studied, for example, in [12,13]. Unique data about the hydrodynamics of drops was obtained in the research. However, it is clear that the hysteresis of contact angles in static conditions on the horizontal surfaces is absent. Thus, the hysteresis in [12] appeared evidently in a dynamic mode, for example, when the drop volume was changed by means of a syringe, as recommended in [2-4]. The hysteresis of contact angles in [13] was not studied at all.

Recently, the states of liquid drops sitting on a horizontal elastic cyclically stretched and then weakened substrate were studied experimentally [14]. A thin rubber band was used as a substrate, and glycerin was used as a liquid. A multi-branch hysteresis of the drop states was detected. The number of branches in the hysteresis can be adjusted by changing the time program for stretching-–loosening the substrate tension.

Thus, the aim of this work is to study a behavior of liquid sessile drops on a stretchable inclined elastic substrate to find out peculiarities of the contact angles hysteresis in the conditions when the drop volume is permanent.

2. Materials and methods

A rubber band was chosen as an elastic substrate for research (we used a famous rubber Martens' bandage; bands of necessary sizes were cut from it). Each band was of the following sizes: the length was 15 cm, the width was 1.5 cm, and the thickness was 0.5 mm. The band had a smooth work surface with less than 100 µm roughness. (Fig. 1).

A 46% water glycerol solution was chosen as a liquid, the producer was OJSC “Samaramedprom” (Russia) (hereinafter referred to as the “glycerin”). The glycerin was slightly colored to improve the visualization quality. It is known that glycerin is a weakly volatile liquid. Thus, the change of the drop volume during the experiment (less than 1 min) can be neglected.

 

Fig. 1. Micro image of the work surface of a nonstretched rubber band (scale bar is 100 mm) obtained by means of the optical digital microscope ‘Levenhuk’ (MODEL D50L NG) [15].

Glycerin does not wet this rubber sample. The result of the contact angle measurement in the glycerin drop on the nonstretched horizontal rubber substrate is shown in Fig. 2. It is 99⁰.

The main experiments were carried out in the following way. A mark of a specified length was drawn on the band edge. It was possible to control the band elongation with the help of this mark. The band was placed on the plane inclined table before the experiment. One of the band ends was fixed, and then the band was stretched along the table surface until the elongation of δ ≥ 2. After that, a glycerin drop was placed on the work surface of the stretched band near the mark. Then, the force holding the band in the stretched condition was smoothly decreasing up to zero during 7‒8 s (the 1^st step); some seconds left, the band again was stretched up to the elongation of δ ≥ 2 (the 2^nd step); and finally, the band was completely free from the stretching force (the 3^rd step).

 

Fig. 2. The results of the contact angle measurement in the glycerin drop located on the horizontal rubber surface; the rubber band is not stretched (obtained by means of the optical digital microscope ‘Celestron’ (MODEL #44302-A) [15]).

 

The dynamics of the drop sliding deformation was recorded by means of the optical digital microscope ‘Celestron’ (MODEL #44302-A) [15] in the video recording mode with the frequency of 20 fps. The following values were measured on each frame: the mark size, the length of the drop base l, and the angles θ_A and θ_R (Fig. 3). The variables l, θ_A, and θ_R characterize the drop state.

The velocity of the mark relative length change was not more than 0.1 s^-1 on each step (the 1^st ‒ 3^rd) in the experiments. Thus, the measurement of the drop geometry parameters l, θ_A, and θ_R can be considered quasi-stationary; the Landau- Levich effects of the liquid drag by means of the moving substrate [16,17] can be neglected.

All the experiments were carried out at the room temperature of 18⁰ Ñ and normal air pressure of 750 Torr.

 

Fig. 3. The scheme of the drop located on the inclined plane; the measured values are explained here.

3. Experimental results

After the described experiment had been repeated several times, the following typical drop behavior was noticed. The form, the base sizes and the contact angles are changed slightly; the drop itself is as if sliding on the squeezed band (a sliding mode) during the 1^st step. The drop elongates together with the band when the band is stretched on the 2^nd step. When the band is again squeezed, the drop base is also decreased on the 3^rd step. The drop edge remains as if stuck to the band surface along the whole perimeter on the 2^nd and 3^rd steps (a pinning mode).

The diagram of the dependence of the drop base relative length λ = l / l _A and the band elongation d is presented in Fig. 4. Here, l _A is an initial size of the drop base corresponding to the A point on the diagram. Fig. 4 illustrates the drop behavior. The diagram is constructed for one of the experiments; the table slope angle is 20⁰, and the values of the diagram points are obtained after à scene selection of the video recording and frames analysis. The characteristic points B, C and D are also stated in the diagram; they are close to the ends of the 1^st, 2^nd, and 3^rd steps, respectively.The roughnesses of curves on the diagram are connected with unremovable pixel errors of images.

There are two strait dotted lines on the diagram in Fig. 4: a horizontal line of the ideal sliding and an inclined line of the ideal (without sliding) drop drag by means of the moving band. It is evident on the 1^st step that nevertheless the drop is sliding on the band with a little drag, since the diagram is a little above the line of the ideal drag. A complete drop drag by means of the band does not happen on the 2^nd and 3^rd steps since the diagram is a little below the line of the ideal drag and there is a little sliding.

The drop states corresponding to the points A, B, C, and D are shown in video frames in Fig. 5. It is evident that the drop can be in different states (Fig. 5 a,c) having one and the same value of the band elongation d; it depends on its prehistory. Thus, a new hydrophysical effect was found in this research. It is hysteresis of the drop states on the inclined elastic stretchable substrate.

We carried out several experiments in which the 2^nd and 3^rd steps of stretching and squeezing were repeated with one and the same drop many times. Finally, it was defined that the drop state passes along the diagram line between the C and D points in Fig. 4 many times, and the drop state passes between the A and B points only once, on the 1^st step.

We also carried out several experiments without the 1^st step, i.e. the drop was placed on the nonstretched band, and the hysteresis of states did not happen. It means that the drop motion on the 1^st step has a special character, at which the drop drag by means of the squeezed band is practically absent.

 

Fig. 4. The diagram of the dependence of the drop base relative length λ = l / l _A and the band elongation d. Points A, B, C, and D correspond to the start of the 1^st, 2^nd, and 3^rd steps and some frames from the video recording in Fig. 5; the numbered arrows show the start of the corresponding steps; the horizontal dotted line is a line of the ideal sliding, the inclined dotted line is a line of the ideal drop drag (without sliding) by means of the moving band.

 

The diagrams of the dependence of the contact angles θ_A and θ_R and the band elongation d are presented in Fig. 6. They illustrate the hysteresis evolution of contact angles at the band deformation. It is evident that θ_A is below the conditional boundary dividing the areas of wettability ( θ_A < 90⁰) and non-wettability (θ_A > 90⁰) at the initial moment, while θ_R is above this boundary. θ_A grows on the 1^st step and exceeds the level of 90⁰ at the end of the step, and θ_R is almost stable. Both contact angles θ_A and θ_R cross the level of 90⁰ on the next 2^nd and 3^rd steps.These crossings correspond to the transition from wettability to non-wettability and vice versa.

A special character of the 1^st step is also evident here, it corresponds to the found hysteresis of states; besides, this hysteresis of states happens together with a famous hysteresis of contact angles.

 

Fig. 5. The drop states corresponding to the steps: a) – the initial state of the drop (point A in Fig. 4); b) – the drop state at the end of the 1^st step(point B); c) – the drop state at the end of the 2^nd step(point C), d) – the drop state at the end of the 3^rd step (point D), the place and the length of the mark are evident under the drop base; the images are obtained by the optical digital microscope ‘Celestron’ (MODEL #44302-A) [14].

4. Conclusions

The authors studied the states of liquid sessile drops on a stretchable inclined elastic substrate to find out peculiarities of the contact angles hysteresis.

The authors carried out the experiments in which the drops volume remained stable in every experiment. A special weakly volatile liquid – glycerin was used for this purpose. A rubber band was used as a substrate.

The band was placed on the plane inclined table before every experiment. One of the band ends was fixed, and then the band was stretched along the table surface up to the specified elongation. A glycerin drop was placed on the work surface of the stretched band. After that, the force holding the band in the stretched condition was smoothly decreasing up to zero (the 1^st step); then some seconds left, the band was stretched again (the 2^nd step), and finally, the band was completely free from the stretching force (the 3^rd step).

The drop elongation and the contact angles were recorded. An anomalous behavior of the drop was found on the 1^st step during which the measured drop parameters remained unchanged. Finally, a new hydrophysical effect was found. It is hysteresis of the drop states which happens together with a famous hysteresis of contact angles.

 

Fig. 6. The diagrams of dependence of the contact angles θ_A and θ_R and the band elongation d. Points A, B, C, and D correspond to the start of the 1^st, 2^nd, and 3^rd steps and some frames from the video recording in Fig. 5; the numbered arrows show the start of the corresponding steps; the horizontal dotted line is a conditional boundary dividing the areas of wettability (θ_A,R < 90⁰) andnon-wettability (θ_A,R > 90⁰).

References

[1] Israelchvili J. Intermolecular and surface forces. Oxford: Elsevier, 1991.

[2] de Gennes P.-G., Brochard-Wyart F., and Quéré D. Capillarity and wetting phenomena. Drops, bubbles, pearls, waves. NY: Springer- Verlag, 2004.

[3] Bormashenko E. Yu. Wetting of real surfaces. Berlin: Walter de Gruyter GmbH, 2013.

[4] Butt H.-J., Liu J., Koynov K., Straub B., Hinduja C., Roismann I., Berger R., Li X., Vollmer D., Steffen W., and Kappl M. Contact angle hysteresis / Curr. Opin. in Colloid & Interface Sci. 2022. V. 59. ¹ 1. P. 101574-1-19.

[5] Pierce E., Carmona F. J., and Amirfazli A., Understanding of sliding and contact angle results in tilted plate experiments / Colloids and Surfaces A: Physicochem. Eng. Aspects.

2008. V. 323. ¹ 1-3. P. 73-82.

[6] Ravazzoli P. D., Cuellar I., González A. G., and Diez J. A. Contact-angle-hysteresis effects on a drop sitting on an incline plane / Phys. Rev. E. 2019. V. 99. ¹ 4. P. 043105-1-10.

[7] Dubinov A.E., Kozhayeva J.P., Lyubimtseva V.A., and Selemir V. D. Plasma as a surfactant: A new capillary effect and a new wetting effect induced by nanosecond spark discharges / IEEE Trans. Plasma Sci. 2017. V. 45. ¹ 12. P. 3094-3099.

[8] Dubinov A.E., Iskhakova D.N., and Lyubimtseva V.A. An inversion of contact angle hysteresis when a liquid drop slides up on an inclined plane under the spark discharge action / Phys. Fluids. 2021. V. 33. ¹ 6. P. 061707-1-4.

[9] Liu J.-L. and Feng X.-Q. On elastocapillarity: A review / Acta Mechanica Sinica. 2012. V. 28. ¹ 4. P. 928-940.

[10] Style R. W., Jagota A., Hui C.-Y., and Dufresne E. R. Elastocapillarity: Surface tension and the mechanics of soft solids / Annu. Rev. Condens. Matter Phys. 2017. V. 8. ¹ 1. P. 99–118.

[11] Bico J., Reyssat E., and Roman B. Elastocapillarity: When surface tension deforms elastic solids / Annu. Rev. Fluid Mech. 2018. V. 50. ¹ 1. P. 629–659.

[12] Cho S. J., Nam H., Ryu H., and Lim G. A rubberlike stretchable fibrous membrane with anti-wettability and gas breathability / Adv. Funct. Mater. 2013. V. 23. ¹ 45. P. 5577–5584.

[13] do Nascimento R. M., Schmitt J.-F., Sarig U., Rodrigues J. E. F. S., Pecharromán C., Ramos A. P., Ciancaglini P., Faita F. L., Rahouadj R., Hernandes A. C., and Bechtold I. H. Surface wettability of a natural rubber composite under stretching: A model to predict cell survival / Langmuir. 2021. V. 37. ¹ 15. P. 4639-4646.

[14] Dekhtyar V. A. and Dubinov A. E. Experimental detection of the multi-branch hysteresis of the droplet states sitting on a horizontal stretchable substrate / Phys. Fluids. 2023. V. 35. ¹ 7. P. 071707-1 ‒ 4.

[15] Dekhtyar V. A. and Dubinov A. E. Visualization of liquids flows in microfluidics and plasma channels in nanosecond spark microdischarges by means of digital microscopy / Scientific Visualization. 2023. V. 15. ¹ 1. P. 1-16.

[16] Landau L. and Levich B. Dragging of a liquid by a moving plate / Acta Physicochimica URSS. 1942. V. 17. ¹ 1-2. P. 141-153.

[17] Mayer H. C. and Krechetnikov R. Landau- Levich flow visualization: Revealing the flow topology responsible for the film thickening phenomena / Phys. Fluids. 2012. V. 24. ¹ 5. P. 052103-1-33.