Dynamics of Impact of polymer droplets on Viscoelastic Surfaces

Dynamics of Impact of polymer droplets on Viscoelastic Surfaces | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Dynamics of Impact of polymer droplets on Viscoelastic Surfaces Saurabh Yadav, Binita Pathak This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4480907/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 24 Sep, 2024 Read the published version in Experiments in Fluids → Version 1 posted 9 You are reading this latest preprint version Abstract Droplet impact on soft surfaces is important in many industrial, biological and agricultural applications. In this paper, we have analysed the dynamics of impact of polymer droplets upon PDMS surfaces. We varied the impact velocity (0.5-2 m/s) and found that impact velocity plays a crucial role in the process. The elasticity of the substrate has also been varied to study its effect upon the droplet dynamics. We delineate the entire process into three different stages and employ force balance equations to identify the governing forces during each stage. The initial spreading is strongly inertia-controlled and the maximum diameter obeys a power-law relation with the Weber number (We. 25 ), irrespective of the impact velocity and the surface properties. The viscoelastic nature of the surface has a dominant influence upon the retraction of the droplets. The effect is more prominent at a higher velocity wherein, the droplet retraction is completely eliminated. A damped harmonic oscillator-type analogy shows that the damping is higher on soft surfaces and at higher velocities. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1 Introduction The study of droplet impact is crucial due to its myriad industrial applications like fuel injection, painting, windshield coatings, aircraft icing prevention, electronic-component welding, fuel combustion optimization, etc. (Worthington 1877 ; Bragg 1996 ; Martin et al. 2008 ; Bolleddula et al. 2010 ; Liu et al. 2013 ; Moghtadernejad et al. 2016 ). Some other applications include pesticide spraying on crops, insecticides spray on fly wings, tissue engineering, to name a few (Gart et al. 2015 ; Zhang et al. 2019 ). In most of these applications, the droplets impact on tissues, leaves, etc. which are soft surfaces. The fate of the droplets which impact on a surface depends upon several factors such as droplet impact velocity, droplet size, surface tension, viscosity, as well as the surface properties (Yarin 2006 ). The impact dynamics on hard surfaces have been rigorously studied due to its fundamental importance and its applications. The major forces governing the process of impact are inertial force, force of surface tension and viscous force. Accordingly, the strength of non-dimensional parameters like the Reynolds number \(\left(Re\right)\) , Weber number \(\left(We\right)\) , among others, determine the different impact outcomes such as deposition, rebound, prompt splash, corona splash, partial rebound, and receding breakup (Rioboo et al. 2001 ; Josserand and Thoroddsen 2016 ). For instance, droplets of low We undergo deformation, deposition and retraction (Rioboo et al. 2002 ; CLANET et al. 2004 ; Yarin 2006 ). On the other hand, high \(We\) impact is dominated by the phenomena of splashing wherein the inertial force dominates over stabilizing surface tension forces(Xu et al. 2005 ; Liu et al. 2010 ) .Surprisingly, a limited work has been done towards the understanding of the impact dynamics on soft/elastic surfaces (Rioboo et al. 2010 ; Chen et al. 2011 ; Alizadeh et al. 2013 ; Weisensee et al. 2016 ; Howland et al. 2016 ). Unlike a solid surface, the dynamics of droplet impact on a soft surface is more complex due to deformation of the surface. The surface deformation alters the contact angle hysteresis depending upon the impact velocity (Rioboo et al. 2010 ). The energy lost due to deformation of soft substrates also reduces the rate of splashing of the droplets (Howland et al. 2016 ). Chen and Li demonstrated that the droplet rebounds more on soft surfaces due to the formation of an air film at the interface (Chen and Li 2010 ). Additionally, a droplet impact on a soft surface undergoes oscillations. Chen et al. gave insights into the oscillations of droplets on a PDMS coated surface by comparing it to a spring mass damper system and different outcome regimes were defined based on the Weber number (Chen et al. 2016 ). These results are related to Newtonian fluids. However, most of the biological and agricultural applications involve non-Newtonian fluids. The behaviour of these droplets is drastically different from the pure fluid droplets. For instance, protein-laden droplets rebounce less even on a hard surface due to high rate of viscous dissipation (Smith and Bertola 2010 ). The droplets alter the wetting properties of the substrate which could also lead to a reduction of rebounce of the droplets (Izbassarov and Muradoglu 2016 ). Such polymer added droplets promote deposition and are highly preferable in applications like spray of pesticide over leaves (Wirth et al. 1991 ; Bergeron et al. 2000 ). The enhanced deposition is attributed to the rheological properties of the fluids which induces resistance to the motion of droplets (Bartolo et al. 2007 ). Vega and Pita showed that addition of small amount of polymer completely suppresses splashing mainly due to elastic energy of the polymer chains (Vega and Castrejón-Pita 2017 ). However, the behaviour of such polymer droplets has not been explored on a soft substrate. Therefore, a systematic approach is required to understand the dynamics of such impinging droplets. In this work, we study the impact of polymer droplets upon soft surfaces. We deployed water based PEO droplets on PDMS surfaces of different softness. We delineate three different stages during the entire droplet lifetime and analysed these stages using major governing forces. The effect of the surface softness and the droplet impact velocities has been elucidated. 2 Preparation of substrates The Polydimethylsiloxane (PDMS) surfaces were prepared using a 2-component silicone fluid (SYLGARD 184, Dow Corning, Wiesbaden, Germany). The curing agent and the base (silicone elastomer) were taken in specific weight ratios (1:10, 1:20, or 1:30). Both the base and the curing agent were mixed properly using a stirrer to form a homogenous solution. The mixture was degassed in a desiccator using a vacuum pump for around 30 minutes to eliminate trapped air bubbles. The degassed mixture was then transferred into Petri dishes and cured in an oven at 100°C for 10 hours. This curing step triggers crosslinking between the base and curing agent, forming the final elastomeric PDMS surfaces. The weight ratio of the silicone elastomer to the curing agent plays a crucial role in determining the mechanical properties. By adjusting the weight ratio, which controls the "crosslinking density" of the final PDMS network, its stiffness was manipulated. A higher base-to-curative agent ratio translated to a softer and more compliant elastomer, while a lower ratio yielded a stiffer and more rigid surface. The PDMS surfaces obtained were about ~ 4.98 mm in thickness. 3 Experimental setup The experimental setup consists of a syringe pump to generate the droplets of about 1 mm radius (R 0 ) (Fig. 1 ). The dynamics of droplet impact on the PDMS surfaces were recorded at 6000 fps using a high-speed camera (Photron Fastcam Mini AX100) equipped with a zoom lens setup (Navitar). The droplets were back illuminated using a DC light source (Veritas Constellation 120e). The surfaces were mounted on a XYZ translational stage for the precise control of the movements. The entire set up was placed on an optical table to avoid any external vibration. The PEO solution was prepared in DI water and the concentration was varied from 0.1 g/L to 1 g/L (of PEO). The dynamics were studied for different velocities (0.5 m/s − 2 m/s) of the droplets. The base case considered was DI water. We have studied the variation of surfaces from the hardest 1:10 (shear modulus, G ~ 500 kPa) to the softest 1:30 (shear modulus, G ~ 70 kPa). 4 Results and discussion 4.1 Dynamics of droplet impact The dynamics of droplet impact on the PDMS surfaces are shown in Fig. 2 . The radius of the droplet was R d0 ~ 1 mm in all the cases. The size is smaller than the capillary lengthscale \(\left(\sqrt{\text{σ/ρg}}\right)\) such that the effect of gravity is neglected in the current study. The entire process can be described in three different stages, namely, spreading, retraction, and oscillation stages, respectively. The spreading phase is identified as an increase in the contact diameter (Fig. 3 ). As the droplet spreads due to its inertia, the PDMS surface hinders the spreading and the fluid accumulates in the peripheral region, thereby forming a thick rim-type structure. The thickness of the rim (h rim ) increases (by about 37%) with an increase in impact velocity (by two times) of the droplet. This is attributed to higher kinetic energy \(\left({\text{E}}_{\text{k}}\text{ ~ }\raisebox{1ex}{$\text{2}$}\!\left/ \!\raisebox{-1ex}{$\text{3}$}\right.\text{π}\text{ρ}{{\text{R}}_{\text{d0}}}^{\text{3}}{\text{v}}^{\text{2}}\right)\) (here, v and ρ are the velocity and the density of the droplet respectively) which overcomes the restriction and spreads more, thereby forming thinner rims. The rate of spreading is slightly reduced (~ 15%) as the concentration of PEO increases from 0–1 g/l due to viscous dissipation. The change in rim thickness (h rim ) is also negligible due to an increase in the PEO concentration (0–1 g/L) which reveals that this phase is mostly inertia-dominated. The spreading phase is also accompanied by the initiation of small scale capillary waves (ripples) at the air-liquid interface (Fig. 2 ). The capillary waves are formed on the peripheral rim and progress radially inward. The waves are dampened with an increase in the concentration of PEO due to high viscous dissipation. As the droplet attains the maximum contact diameter on the substrate, the surface energy of the droplet causes it to retract (Fig. 2 ). The droplet retracts to a top hat-type structure with the maximum height (H max ) attained in the entire droplet lifetime. The retraction phase ends with the pining of the contact line at the equilibrium diamter. The top hat structure eventually collapses into damped oscillations, thereby, dissipating the residual kinetic energy (oscillation phase in Fig. 2 ). The process ends as the droplet attains the equilibrium state. 4.2 Spreading Phase The spreading phase is governed by inertia, surface energy and viscous forces. The maximum diameter attained at the end of the spreading phase (D max ) is only marginally reduced on the softer surface (1:30) as compared to the surface 1:10 (Fig. 3 ). This indicates that the influence of the surface viscoelasticity is not considerable upon the spreading of the droplets. Alternatively, the D max increases considerably (~ 25%) with a 2-fold rise in the impact velocity. The rise in the rate of spreading \(\left(\frac{\text{dD}}{\text{dt}}\right)\) with the velocity is clearly depicted in the temporal variation of the contact line \(\left(\raisebox{1ex}{$\text{D}$}\!\left/ \!\raisebox{-1ex}{${\text{D}}_{\text{0}}$}\right.\right)\) of the droplets plotted in Fig. 3 a. It can thus be inferred that the inertia plays a dominant role in the spreading phase. The role of viscosity can be observed by comparing the D max with that of the base fluid. The rate of spreading for 1% PEO droplets is only marginally slower than water droplets (~ 10%) for the same impact velocity on similar surfaces (Fig. S1 in supplementary material). Accordingly, the maximum diameter attained at the end of the phase is only slightly reduced for the polymer droplets. The viscous dissipation energy \(\left({\text{E}}_{\text{v}}\text{ ~ }\mu \frac{{V}_{0}}{{l}_{v}}\forall \right)\) [here, l v is the thickness of the droplet of volume ( \(\text{∀}\text{)}\) at maximum diameter which is obtained by mass balance during the spreading phase. \(\mu\) is the fluid viscosity] is much smaller (~ 3 orders) than the inertia of the droplets \(\left({\text{E}}_{\text{k}}\right)\) . Therefore, the effect of viscosity is not appreciable during the spreading phase. However, the E v increases by an order of magnitude (10 − 7 J) towards the end of the spreading of the droplets. Therefore, the total spreading timescale is less than the inertial timescale \(\left(\frac{{\text{D}}_{\text{0}}}{{\text{V}}_{\text{0}}}\right)\) as some energy is lost due to the viscous dissipation. The scaling of the force balance equation along with the mass balance gives: \(\left(\frac{{\text{D}}_{\text{max}}}{{\text{D}}_{\text{0}}}\text{ ~}{\text{ We}}^{\text{1/4}}\right)\) (CLANET et al. 2004 ) This scaling is validated with the experimental data \(\left(\frac{{\text{D}}_{\text{max}}}{{\text{D}}_{\text{0}}}\right)\) plotted for different cases which collapses to a single curve (Fig. 3 b). 4.3 Retraction Phase The droplets stay at the maximum diameter (D max ) for some time (~ 5 ms) prior to the retraction (Fig. 3 a). As the inertia of the droplet reduces drastically due to spreading, the surface energy \(\left({\text{E}}_{\text{s}}\text{~ πD}\text{h}\text{σ}\right)\) (here, \(\text{σ}\) is the coefficient of surface tension of the droplet and \(\text{h }\text{=}\frac{\text{2}}{\text{3}}\frac{{{\text{D}}_{\text{0}}}^{\text{3}}}{{\text{D}}^{\text{2}}}\) is the thickness) increases by about an order of magnitude (Fig. 4 a). Therefore, the droplet tends to retract to minimize the total energy. The retraction of the droplets ends by pining of the diameter at the equilibrium position (D eq ). As noted earlier during spreading, an increase in the impact velocity (by 2-times) leads to an increased D max . Larger contact diameter possess higher surface energy (E s increases by ~ 2 times). Increased velocity also causes an increase in the viscous dissipation \(\left({E}_{v}\sim\mu \frac{{V}_{0}}{{l}_{v}}\forall \right)\) by ~ 4 times. However, the rate of viscous dissipation is smaller than the surface energy of the droplets (Fig. 4 a). Therefore, increased velocity causes a net rise (~ 35%) in the rate of retraction (Fig. 4 b). On the contrary, the increase in surface softness shows a reverse effect upon retraction. The maximum diameter attained at the end of the spreading stage (D max ) is seen to be slightly reduced on the soft surface (1:30). Therefore, the surface energy of the surface is also less as compared to the hard one (1:10). Accordingly, the retraction rate is reduced (~ 50%) on soft surfaces. Moreover, the soft surfaces deform under the impact of the droplets. The deformation leads to the formation of a wetting ridge near the contact line which also restricts the retraction of the droplet. The deformation ( \(\delta\) ) of soft surface (1:30) is quite high at higher velocity ( \({\delta }_{2m/s} \sim\) 2 orders as compared to that at 0.5 m/s). Therefore, it causes a strong pining of the contact line (at D max ) on the surface. Consequently, no retraction was observed on this surface at 2 m/s. A balance of the surface tension and viscous dissipation forces gives a timescale of \(\frac{\text{μ}{\text{D}}_{\text{max}}}{\text{σ}}\) . The timescale is validated with the experimental data during the retraction phase (Fig. 4 b). The potential energy (PE) due to deformation of the surface has not been considered in the scaling analysis as the PE is about an order of magnitude lower than the viscous dissipation (Fig.S2 in the supplementary material). However, the PE is comparable to the viscous dissipation at high velocity (2 m/s). Therefore, the scaling is not valid in this case. 4.4 Oscillation phase The retraction phase is followed by oscillation of the droplets. The droplet with pinned contact line behaves like a damped harmonic oscillator (Fig. 5 a). The oscillations are dampened due to dissipation of energy on the surface. The energy is dissipated as viscous dissipation within the boundary layer. The rate of viscous dissipation \(\left(\dot{\text{∅}}\text{ ~}{\int }_{\text{0}}^{{\text{R}}_{\text{d}}}\frac{\text{2πrμ}}{\text{δ}}{\text{v}}^{\text{2}}\text{dr}\right)\) (here, \(\text{δ}\) is boundary layer thickness which is approximated as the thickness of the rim in the present work. µ is the viscosity of the droplets) increases by about an order of magnitude for an increase in the velocity by two times. Therefore, the oscillations are dampened faster for the droplets with high impact velocity. Accordingly, the total time of oscillation (t os ) is also reduced (Fig. 5 b (closed symbols)). For instance, t os,.5 m/s ~ 2.5 t os,1m/s while t os,1 m/s ~ 3 t os,2m/s . Also, maximum energy of the droplet impacting at a high velocity is utilized in spreading of the droplet to form a larger contact diameter. The residual energy is thus not sufficient for oscillation of these droplets, which is clearly identified as highly dampened at a velocity of 2 m/s in Fig. 5 a. The reduction in residual energy is also evident as the initial peak height (h po ) reduces by ~ 3 times for a 2-fold rise in the velocity (Fig. S3 in supplementary material). Furthermore, the pinned contact diameter is larger for droplets impacting at a higher velocity. The wavelength of oscillation is longer on a larger contact diameter and thus the frequency is low which is depicted by a gradual decrease in the frequency with the impact velocity (Fig. 5 b). The oscillations can be expressed by the equation of a typical spring mass damper system: \(\text{m}\ddot{\text{h}}\text{+c}\dot{\text{h}}\text{+}\text{kh}\text{=0}\) (here, m is mass, c is the damping coefficient and k is the spring constant. h represants the displacement and it is the droplet height in this case). The solution of this equation gives the displacement of the system about its equilibrium position as: \(\text{h= }{\text{h}}_{\text{o}}\text{+(}{\text{c}}_{\text{1}}{\text{sin}\left(\text{ωt}\right)\text{+c}}_{\text{2}}\text{cos}\text{⁡(}\text{ωt}\text{))}{\text{e}}^{\text{-}\text{ξt}}\) . The oscillation frequency \(\text{⁡(}\text{ω) }\) is estimated from peak to peak time and the viscous damping factor, \(\text{ξ=}\frac{c}{2m}\) is evaluated by fitting a curve to the experimental data. The ‘c’ estimated from the experimental data is of order ~ 10 − 5 Ns/m for the impact velocity of 0.5 m/s and it increases by 2–3 times for an increase in the impact velocity upto 2 m/s. The rise in the damping coefficient with an increase in the velocity is due to the higher rate of energy dissipation \(\left(\dot{\text{∅}}\text{ ~}{\int }_{\text{0}}^{{\text{R}}_{\text{d}}}\frac{\text{2πrμ}}{\text{δ}}{\text{v}}^{\text{2}}\text{dr}\right)\) . Furthermore, ‘c’ increases slightly (~ 15%) on softer surfaces. The increase is due to viscoelastic behavior of the softer surface. The softer surface (1:30) deforms about 25 times more as compared to the surface 1:10 which pins the droplet at a larger D eq than that on the harder one. Accordingly, the thickness of the layer for dissipation of energy \((\text{δ)}\) is smaller as the droplet volume is constant and thus the value of c which is related to energy dissipation ( \(\dot{\text{∅}}\) ) is large. The total timescale of oscillation is also slightly reduced (~ 10%) on soft surface (1:30) as compared to a harder surface (1:10) (Fig. 5 b). The spring constant is estimated as \(\text{k = m}\left({\text{ω}}^{\text{2}}\text{+}{\left(\frac{\text{c}}{\text{2m}}\right)}^{\text{2}}\right)\) and it signifies the tendency of the system to restore its energy (Chen et al. 2016 ). k decreases by 75–80 % for an increasein the impact velocity from 0.5 m/s – 2 m/s. Since there are no external forces to drive the oscillations, the surface energy which is the only restoration energy of the system would dissipate with time. Since the dissipation is high for higher velocities, the restoration energy is also reduced. 5 Conclusions In this work, we experimentally investigated the impact of polymer droplets on viscoelastic surfaces. The complex dynamics of impact can be identified as three stages, namely, spreading, retraction and oscillation. The spreading of droplets is mainly governed by the inertia of the droplets. A scaling of the inertia and the surface tension forces gives an estimation of the maximum contact diameter formed during spreading as a function of the Weber number as \(\left(\frac{{\text{D}}_{\text{max}}}{{\text{D}}_{\text{0}}}\text{ ~}{\text{ We}}^{\text{1/4}}\right)\) . The spreading stage is followed by the retraction of the droplets. The droplet retracts to minimize its surface energy. Softer surfaces restrict the movement of the droplet, thereby reducing the rate of retraction as well. The retraction stage is completely eliminated on softer surface at a higher velocity (2 m/s) due to large deformation which pins the droplet at the maximum contact diameter. The end of the retraction stage is identified by pining of the contact diameter in all the cases. The pinned droplet finally oscillates like a damped oscillator till the equilibrium state. The oscillations dampen at a higher rate at high impact velocity due to high viscous dissipation rate and less energy available for the oscillations. A simple spring-mas-damper system analogy also shows that the damping coefficient which is related to energy dissipation is higher on softer surfaces due to its viscoelastic behavior. Declarations The authors declare no financial or personal interests to influence the reported work. *Corresponding author Binita Pathak Department of Mechanical Engineering Indian Institute of Technology, IIT BHU Varanasi Varanasi-221005, India Email: [email protected] [email protected] Tel: +91-8105431120 References Alizadeh A, Bahadur V, Shang W, et al (2013) Influence of Substrate Elasticity on Droplet Impact Dynamics. Langmuir 29:4520–4524. https://doi.org/10.1021/la304767t Bartolo D, Boudaoud A, Narcy G, Bonn D (2007) Dynamics of Non-Newtonian Droplets. Phys Rev Lett 99:174502. https://doi.org/10.1103/PhysRevLett.99.174502 Bergeron V, Bonn D, Martin JY, Vovelle L (2000) Controlling droplet deposition with polymer additives. Nature 405:772–775. https://doi.org/10.1038/35015525 Bolleddula DA, Berchielli A, Aliseda A (2010) Impact of a heterogeneous liquid droplet on a dry surface: Application to the pharmaceutical industry. 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Supplementary Files SUPPLEMENTARYMATERIAL.docx Cite Share Download PDF Status: Published Journal Publication published 24 Sep, 2024 Read the published version in Experiments in Fluids → Version 1 posted Editorial decision: Revision requested 26 Aug, 2024 Reviews received at journal 18 Jun, 2024 Reviewers agreed at journal 12 Jun, 2024 Reviewers agreed at journal 12 Jun, 2024 Reviewers agreed at journal 12 Jun, 2024 Reviewers invited by journal 12 Jun, 2024 Editor assigned by journal 28 May, 2024 Submission checks completed at journal 27 May, 2024 First submitted to journal 26 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4480907","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":310762745,"identity":"95919c83-14a8-4d12-b62d-4c97450a1bd2","order_by":0,"name":"Saurabh Yadav","email":"","orcid":"","institution":"Indian Institute of Technology, IIT-BHU Varanasi","correspondingAuthor":false,"prefix":"","firstName":"Saurabh","middleName":"","lastName":"Yadav","suffix":""},{"id":310762748,"identity":"804e9594-5775-4304-8e9c-e6800c6ea47a","order_by":1,"name":"Binita Pathak","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVRIie3SIQvCQBTA8SfCVk5Xr/gdDhaG4Dex7BBMXlcwDAaziFb9FsY13zgwnd1gcAgmwyyCRZycaHIbJsP9w/HC+3EXDsBk+sNYHQD1WEesBa/ZLyN6z/IxqUSeh94jDN6kKM9uJDIbQ9ex1UVe4j0PbHmANP5O2mHTR9yAWE7FChN14gHpM+Cq4GGSMEQLxGrXyEkkeQADAB6VkTuI9Y4cNHHOFUgS5bdQAprQklvaYU62MyoWqp8PSroRPTEsIp6j3Gx47Yj5RB6zUSxbc6eXprcC8op+Rgve/8FkMplMv/YA9Pdg13woXw0AAAAASUVORK5CYII=","orcid":"","institution":"Indian Institute of Technology, IIT-BHU Varanasi","correspondingAuthor":true,"prefix":"","firstName":"Binita","middleName":"","lastName":"Pathak","suffix":""}],"badges":[],"createdAt":"2024-05-26 17:08:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4480907/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4480907/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00348-024-03886-x","type":"published","date":"2024-09-24T15:57:42+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":57917165,"identity":"d8559034-b7b3-4b5d-b396-b9dd7fdca5a2","added_by":"auto","created_at":"2024-06-07 12:24:49","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":97391,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic of the experimental set up.\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/189c6796842a515e080d6d23.png"},{"id":57917169,"identity":"0871d9c7-e260-413c-8273-d67fec3ac880","added_by":"auto","created_at":"2024-06-07 12:24:50","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":551245,"visible":true,"origin":"","legend":"\u003cp\u003eThe impact dynamics of (a) DI droplet and (b) 1 g/l PEO droplet on PDMS surface (1:30).\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/fb62460483a832fff421f885.png"},{"id":57917170,"identity":"8a0b7e5a-b3d6-4cf3-995a-7433aa541006","added_by":"auto","created_at":"2024-06-07 12:24:50","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":221274,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The temporal variation of the contact diameter (D) which are non-dimensionalised by the initial droplet diameter (D\u003csub\u003e0\u003c/sub\u003e) [the open symbols represent the values at the impact velocity of 1 m/s and the closed symbols represent the values at the impact velocity of 0.5 m/s respectively], (b) the maximum contact diameter (D\u003csub\u003emax\u003c/sub\u003e) formed at the end of the spreading stage plotted against the We\u003csup\u003e.25\u003c/sup\u003e (We: Weber number).\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/ae05f38e31bdf3c1ce0dcc39.png"},{"id":57917633,"identity":"d4f33548-e81e-4573-819b-78d6d192d3b0","added_by":"auto","created_at":"2024-06-07 12:32:50","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":289361,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/75aa18cca767fda269de0367.png"},{"id":57917167,"identity":"36f54807-9773-44bc-9d7f-626a34b343de","added_by":"auto","created_at":"2024-06-07 12:24:50","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":265716,"visible":true,"origin":"","legend":"\u003cp\u003e(a) The temporal variation of the droplet height (H) non-dimensionalised by the initial droplet diameter (D\u003csub\u003e0\u003c/sub\u003e) for different velocities on the 1:30 surface and (b) the droplet oscillation frequency (ω) [open symbols] and the total oscillation time (t\u003csub\u003eos\u003c/sub\u003e) [closed symbols] for different impact velocities on different surfaces.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/4f01d10f76429c783a866a1f.png"},{"id":65627288,"identity":"c9dac2b9-2de3-4f87-850f-9ce8774d28d8","added_by":"auto","created_at":"2024-09-30 16:14:19","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1738392,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/a59ddda6-4036-49e5-a299-0f47f8e23e64.pdf"},{"id":57917166,"identity":"6bcb71dc-e138-4608-ac77-5e9de9c9758b","added_by":"auto","created_at":"2024-06-07 12:24:49","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":348251,"visible":true,"origin":"","legend":"","description":"","filename":"SUPPLEMENTARYMATERIAL.docx","url":"https://assets-eu.researchsquare.com/files/rs-4480907/v1/8c5c943386b197b95f962dd0.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Dynamics of Impact of polymer droplets on Viscoelastic Surfaces","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe study of droplet impact is crucial due to its myriad industrial applications like fuel injection, painting, windshield coatings, aircraft icing prevention, electronic-component welding, fuel combustion optimization, etc. (Worthington \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1877\u003c/span\u003e; Bragg \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Martin et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Bolleddula et al. \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Moghtadernejad et al. \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Some other applications include pesticide spraying on crops, insecticides spray on fly wings, tissue engineering, to name a few (Gart et al. \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Zhang et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). In most of these applications, the droplets impact on tissues, leaves, etc. which are soft surfaces. The fate of the droplets which impact on a surface depends upon several factors such as droplet impact velocity, droplet size, surface tension, viscosity, as well as the surface properties (Yarin \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). The impact dynamics on hard surfaces have been rigorously studied due to its fundamental importance and its applications. The major forces governing the process of impact are inertial force, force of surface tension and viscous force. Accordingly, the strength of non-dimensional parameters like the Reynolds number \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(Re\\right)\\)\u003c/span\u003e\u003c/span\u003e, Weber number \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(We\\right)\\)\u003c/span\u003e\u003c/span\u003e, among others, determine the different impact outcomes such as deposition, rebound, prompt splash, corona splash, partial rebound, and receding breakup (Rioboo et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Josserand and Thoroddsen \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). For instance, droplets of low \u003cem\u003eWe\u003c/em\u003e undergo deformation, deposition and retraction (Rioboo et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; CLANET et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Yarin \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2006\u003c/span\u003e). On the other hand, high \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(We\\)\u003c/span\u003e\u003c/span\u003e impact is dominated by the phenomena of splashing wherein the inertial force dominates over stabilizing surface tension forces(Xu et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Liu et al. \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) .Surprisingly, a limited work has been done towards the understanding of the impact dynamics on soft/elastic surfaces (Rioboo et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Chen et al. \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Alizadeh et al. \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Weisensee et al. \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Howland et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eUnlike a solid surface, the dynamics of droplet impact on a soft surface is more complex due to deformation of the surface. The surface deformation alters the contact angle hysteresis depending upon the impact velocity (Rioboo et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). The energy lost due to deformation of soft substrates also reduces the rate of splashing of the droplets (Howland et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Chen and Li demonstrated that the droplet rebounds more on soft surfaces due to the formation of an air film at the interface (Chen and Li \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). Additionally, a droplet impact on a soft surface undergoes oscillations. Chen et al. gave insights into the oscillations of droplets on a PDMS coated surface by comparing it to a spring mass damper system and different outcome regimes were defined based on the Weber number (Chen et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). These results are related to Newtonian fluids. However, most of the biological and agricultural applications involve non-Newtonian fluids. The behaviour of these droplets is drastically different from the pure fluid droplets. For instance, protein-laden droplets rebounce less even on a hard surface due to high rate of viscous dissipation (Smith and Bertola \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2010\u003c/span\u003e). The droplets alter the wetting properties of the substrate which could also lead to a reduction of rebounce of the droplets (Izbassarov and Muradoglu \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Such polymer added droplets promote deposition and are highly preferable in applications like spray of pesticide over leaves (Wirth et al. \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e1991\u003c/span\u003e; Bergeron et al. \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). The enhanced deposition is attributed to the rheological properties of the fluids which induces resistance to the motion of droplets (Bartolo et al. \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Vega and Pita showed that addition of small amount of polymer completely suppresses splashing mainly due to elastic energy of the polymer chains (Vega and Castrej\u0026oacute;n-Pita \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). However, the behaviour of such polymer droplets has not been explored on a soft substrate. Therefore, a systematic approach is required to understand the dynamics of such impinging droplets.\u003c/p\u003e \u003cp\u003eIn this work, we study the impact of polymer droplets upon soft surfaces. We deployed water based PEO droplets on PDMS surfaces of different softness. We delineate three different stages during the entire droplet lifetime and analysed these stages using major governing forces. The effect of the surface softness and the droplet impact velocities has been elucidated.\u003c/p\u003e"},{"header":"2 Preparation of substrates","content":"\u003cp\u003eThe Polydimethylsiloxane (PDMS) surfaces were prepared using a 2-component silicone fluid (SYLGARD 184, Dow Corning, Wiesbaden, Germany). The curing agent and the base (silicone elastomer) were taken in specific weight ratios (1:10, 1:20, or 1:30). Both the base and the curing agent were mixed properly using a stirrer to form a homogenous solution. The mixture was degassed in a desiccator using a vacuum pump for around 30 minutes to eliminate trapped air bubbles. The degassed mixture was then transferred into Petri dishes and cured in an oven at 100\u0026deg;C for 10 hours. This curing step triggers crosslinking between the base and curing agent, forming the final elastomeric PDMS surfaces. The weight ratio of the silicone elastomer to the curing agent plays a crucial role in determining the mechanical properties. By adjusting the weight ratio, which controls the \"crosslinking density\" of the final PDMS network, its stiffness was manipulated. A higher base-to-curative agent ratio translated to a softer and more compliant elastomer, while a lower ratio yielded a stiffer and more rigid surface. The PDMS surfaces obtained were about\u0026thinsp;~\u0026thinsp;4.98 mm in thickness.\u003c/p\u003e"},{"header":"3 Experimental setup","content":"\u003cp\u003eThe experimental setup consists of a syringe pump to generate the droplets of about 1 mm radius (R\u003csub\u003e0\u003c/sub\u003e) (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The dynamics of droplet impact on the PDMS surfaces were recorded at 6000 fps using a high-speed camera (Photron Fastcam Mini AX100) equipped with a zoom lens setup (Navitar). The droplets were back illuminated using a DC light source (Veritas Constellation 120e). The surfaces were mounted on a XYZ translational stage for the precise control of the movements. The entire set up was placed on an optical table to avoid any external vibration. The PEO solution was prepared in DI water and the concentration was varied from 0.1 g/L to 1 g/L (of PEO). The dynamics were studied for different velocities (0.5 m/s \u0026minus;\u0026thinsp;2 m/s) of the droplets. The base case considered was DI water. We have studied the variation of surfaces from the hardest 1:10 (shear modulus, G\u0026thinsp;~\u0026thinsp;500 kPa) to the softest 1:30 (shear modulus, G\u0026thinsp;~\u0026thinsp;70 kPa).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"4 Results and discussion","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Dynamics of droplet impact\u003c/h2\u003e \u003cp\u003eThe dynamics of droplet impact on the PDMS surfaces are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The radius of the droplet was R\u003csub\u003ed0\u003c/sub\u003e ~ 1 mm in all the cases. The size is smaller than the capillary lengthscale \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\sqrt{\\text{\u0026sigma;/\u0026rho;g}}\\right)\\)\u003c/span\u003e\u003c/span\u003e such that the effect of gravity is neglected in the current study. The entire process can be described in three different stages, namely, spreading, retraction, and oscillation stages, respectively. The spreading phase is identified as an increase in the contact diameter (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). As the droplet spreads due to its inertia, the PDMS surface hinders the spreading and the fluid accumulates in the peripheral region, thereby forming a thick rim-type structure. The thickness of the rim (h\u003csub\u003erim\u003c/sub\u003e) increases (by about 37%) with an increase in impact velocity (by two times) of the droplet. This is attributed to higher kinetic energy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({\\text{E}}_{\\text{k}}\\text{ ~ }\\raisebox{1ex}{$\\text{2}$}\\!\\left/ \\!\\raisebox{-1ex}{$\\text{3}$}\\right.\\text{\u0026pi;}\\text{\u0026rho;}{{\\text{R}}_{\\text{d0}}}^{\\text{3}}{\\text{v}}^{\\text{2}}\\right)\\)\u003c/span\u003e\u003c/span\u003e (here, \u003cem\u003ev\u003c/em\u003e and \u003cem\u003eρ\u003c/em\u003e are the velocity and the density of the droplet respectively) which overcomes the restriction and spreads more, thereby forming thinner rims. The rate of spreading is slightly reduced (~\u0026thinsp;15%) as the concentration of PEO increases from 0\u0026ndash;1 g/l due to viscous dissipation. The change in rim thickness (h\u003csub\u003erim\u003c/sub\u003e) is also negligible due to an increase in the PEO concentration (0\u0026ndash;1 g/L) which reveals that this phase is mostly inertia-dominated. The spreading phase is also accompanied by the initiation of small scale capillary waves (ripples) at the air-liquid interface (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The capillary waves are formed on the peripheral rim and progress radially inward. The waves are dampened with an increase in the concentration of PEO due to high viscous dissipation. As the droplet attains the maximum contact diameter on the substrate, the surface energy of the droplet causes it to retract (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The droplet retracts to a top hat-type structure with the maximum height (H\u003csub\u003emax\u003c/sub\u003e) attained in the entire droplet lifetime. The retraction phase ends with the pining of the contact line at the equilibrium diamter. The top hat structure eventually collapses into damped oscillations, thereby, dissipating the residual kinetic energy (oscillation phase in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The process ends as the droplet attains the equilibrium state.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Spreading Phase\u003c/h2\u003e \u003cp\u003eThe spreading phase is governed by inertia, surface energy and viscous forces. The maximum diameter attained at the end of the spreading phase (D\u003csub\u003emax\u003c/sub\u003e) is only marginally reduced on the softer surface (1:30) as compared to the surface 1:10 (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). This indicates that the influence of the surface viscoelasticity is not considerable upon the spreading of the droplets. Alternatively, the D\u003csub\u003emax\u003c/sub\u003e increases considerably (~\u0026thinsp;25%) with a 2-fold rise in the impact velocity. The rise in the rate of spreading \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\frac{\\text{dD}}{\\text{dt}}\\right)\\)\u003c/span\u003e\u003c/span\u003e with the velocity is clearly depicted in the temporal variation of the contact line \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\raisebox{1ex}{$\\text{D}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\text{D}}_{\\text{0}}$}\\right.\\right)\\)\u003c/span\u003e\u003c/span\u003e of the droplets plotted in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea. It can thus be inferred that the inertia plays a dominant role in the spreading phase. The role of viscosity can be observed by comparing the D\u003csub\u003emax\u003c/sub\u003e with that of the base fluid. The rate of spreading for 1% PEO droplets is only marginally slower than water droplets (~\u0026thinsp;10%) for the same impact velocity on similar surfaces (Fig.\u003cspan refid=\"MOESM1\" class=\"InternalRef\"\u003eS1\u003c/span\u003e in supplementary material). Accordingly, the maximum diameter attained at the end of the phase is only slightly reduced for the polymer droplets. The viscous dissipation energy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({\\text{E}}_{\\text{v}}\\text{ ~ }\\mu \\frac{{V}_{0}}{{l}_{v}}\\forall \\right)\\)\u003c/span\u003e\u003c/span\u003e [here, \u003cem\u003el\u003c/em\u003e\u003csub\u003ev\u003c/sub\u003e is the thickness of the droplet of volume (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{\u0026forall;}\\text{)}\\)\u003c/span\u003e\u003c/span\u003eat maximum diameter which is obtained by mass balance during the spreading phase. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu\\)\u003c/span\u003e\u003c/span\u003e is the fluid viscosity] is much smaller (~\u0026thinsp;3 orders) than the inertia of the droplets \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({\\text{E}}_{\\text{k}}\\right)\\)\u003c/span\u003e\u003c/span\u003e. Therefore, the effect of viscosity is not appreciable during the spreading phase. However, the E\u003csub\u003ev\u003c/sub\u003e increases by an order of magnitude (10\u003csup\u003e\u0026minus;\u0026thinsp;7\u003c/sup\u003e J) towards the end of the spreading of the droplets. Therefore, the total spreading timescale is less than the inertial timescale \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\frac{{\\text{D}}_{\\text{0}}}{{\\text{V}}_{\\text{0}}}\\right)\\)\u003c/span\u003e\u003c/span\u003e as some energy is lost due to the viscous dissipation. The scaling of the force balance equation along with the mass balance gives: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\frac{{\\text{D}}_{\\text{max}}}{{\\text{D}}_{\\text{0}}}\\text{ ~}{\\text{ We}}^{\\text{1/4}}\\right)\\)\u003c/span\u003e\u003c/span\u003e(CLANET et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2004\u003c/span\u003e) This scaling is validated with the experimental data \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\frac{{\\text{D}}_{\\text{max}}}{{\\text{D}}_{\\text{0}}}\\right)\\)\u003c/span\u003e\u003c/span\u003e plotted for different cases which collapses to a single curve (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Retraction Phase\u003c/h2\u003e \u003cp\u003eThe droplets stay at the maximum diameter (D\u003csub\u003emax\u003c/sub\u003e) for some time (~\u0026thinsp;5 ms) prior to the retraction (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003ea). As the inertia of the droplet reduces drastically due to spreading, the surface energy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({\\text{E}}_{\\text{s}}\\text{~ \u0026pi;D}\\text{h}\\text{\u0026sigma;}\\right)\\)\u003c/span\u003e\u003c/span\u003e (here, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{\u0026sigma;}\\)\u003c/span\u003e\u003c/span\u003e is the coefficient of surface tension of the droplet and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{h }\\text{=}\\frac{\\text{2}}{\\text{3}}\\frac{{{\\text{D}}_{\\text{0}}}^{\\text{3}}}{{\\text{D}}^{\\text{2}}}\\)\u003c/span\u003e\u003c/span\u003e is the thickness) increases by about an order of magnitude (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). Therefore, the droplet tends to retract to minimize the total energy. The retraction of the droplets ends by pining of the diameter at the equilibrium position (D\u003csub\u003eeq\u003c/sub\u003e). As noted earlier during spreading, an increase in the impact velocity (by 2-times) leads to an increased D\u003csub\u003emax\u003c/sub\u003e. Larger contact diameter possess higher surface energy (E\u003csub\u003es\u003c/sub\u003e increases by ~\u0026thinsp;2 times). Increased velocity also causes an increase in the viscous dissipation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({E}_{v}\\sim\\mu \\frac{{V}_{0}}{{l}_{v}}\\forall \\right)\\)\u003c/span\u003e\u003c/span\u003e by ~\u0026thinsp;4 times. However, the rate of viscous dissipation is smaller than the surface energy of the droplets (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). Therefore, increased velocity causes a net rise (~\u0026thinsp;35%) in the rate of retraction (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb). On the contrary, the increase in surface softness shows a reverse effect upon retraction. The maximum diameter attained at the end of the spreading stage (D\u003csub\u003emax\u003c/sub\u003e) is seen to be slightly reduced on the soft surface (1:30). Therefore, the surface energy of the surface is also less as compared to the hard one (1:10). Accordingly, the retraction rate is reduced (~\u0026thinsp;50%) on soft surfaces. Moreover, the soft surfaces deform under the impact of the droplets. The deformation leads to the formation of a wetting ridge near the contact line which also restricts the retraction of the droplet. The deformation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\delta\\)\u003c/span\u003e\u003c/span\u003e) of soft surface (1:30) is quite high at higher velocity (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\delta }_{2m/s} \\sim\\)\u003c/span\u003e\u003c/span\u003e 2 orders as compared to that at 0.5 m/s). Therefore, it causes a strong pining of the contact line (at D\u003csub\u003emax\u003c/sub\u003e) on the surface. Consequently, no retraction was observed on this surface at 2 m/s. A balance of the surface tension and viscous dissipation forces gives a timescale of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{\\text{\u0026mu;}{\\text{D}}_{\\text{max}}}{\\text{\u0026sigma;}}\\)\u003c/span\u003e\u003c/span\u003e. The timescale is validated with the experimental data during the retraction phase (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb). The potential energy (PE) due to deformation of the surface has not been considered in the scaling analysis as the PE is about an order of magnitude lower than the viscous dissipation (Fig.S2 in the supplementary material). However, the PE is comparable to the viscous dissipation at high velocity (2 m/s). Therefore, the scaling is not valid in this case.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e4.4 Oscillation phase\u003c/h2\u003e \u003cp\u003eThe retraction phase is followed by oscillation of the droplets. The droplet with pinned contact line behaves like a damped harmonic oscillator (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea). The oscillations are dampened due to dissipation of energy on the surface. The energy is dissipated as viscous dissipation within the boundary layer. The rate of viscous dissipation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\dot{\\text{\u0026empty;}}\\text{ ~}{\\int }_{\\text{0}}^{{\\text{R}}_{\\text{d}}}\\frac{\\text{2\u0026pi;r\u0026mu;}}{\\text{\u0026delta;}}{\\text{v}}^{\\text{2}}\\text{dr}\\right)\\)\u003c/span\u003e\u003c/span\u003e (here, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{\u0026delta;}\\)\u003c/span\u003e\u003c/span\u003e is boundary layer thickness which is approximated as the thickness of the rim in the present work. \u0026micro; is the viscosity of the droplets) increases by about an order of magnitude for an increase in the velocity by two times. Therefore, the oscillations are dampened faster for the droplets with high impact velocity. Accordingly, the total time of oscillation (t\u003csub\u003eos\u003c/sub\u003e) is also reduced (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb (closed symbols)). For instance, t\u003csub\u003eos,.5 m/s\u003c/sub\u003e ~ 2.5 t\u003csub\u003eos,1m/s\u003c/sub\u003e while t\u003csub\u003eos,1 m/s\u003c/sub\u003e ~ 3 t\u003csub\u003eos,2m/s\u003c/sub\u003e. Also, maximum energy of the droplet impacting at a high velocity is utilized in spreading of the droplet to form a larger contact diameter. The residual energy is thus not sufficient for oscillation of these droplets, which is clearly identified as highly dampened at a velocity of 2 m/s in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea. The reduction in residual energy is also evident as the initial peak height (h\u003csub\u003epo\u003c/sub\u003e) reduces by ~\u0026thinsp;3 times for a 2-fold rise in the velocity (Fig. S3 in supplementary material).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFurthermore, the pinned contact diameter is larger for droplets impacting at a higher velocity. The wavelength of oscillation is longer on a larger contact diameter and thus the frequency is low which is depicted by a gradual decrease in the frequency with the impact velocity (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eThe oscillations can be expressed by the equation of a typical spring mass damper system: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{m}\\ddot{\\text{h}}\\text{+c}\\dot{\\text{h}}\\text{+}\\text{kh}\\text{=0}\\)\u003c/span\u003e\u003c/span\u003e (here, m is mass, c is the damping coefficient and k is the spring constant. h represants the displacement and it is the droplet height in this case). The solution of this equation gives the displacement of the system about its equilibrium position as: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{h= }{\\text{h}}_{\\text{o}}\\text{+(}{\\text{c}}_{\\text{1}}{\\text{sin}\\left(\\text{\u0026omega;t}\\right)\\text{+c}}_{\\text{2}}\\text{cos}\\text{⁡(}\\text{\u0026omega;t}\\text{))}{\\text{e}}^{\\text{-}\\text{\u0026xi;t}}\\)\u003c/span\u003e\u003c/span\u003e. The oscillation frequency \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{⁡(}\\text{\u0026omega;) }\\)\u003c/span\u003e\u003c/span\u003eis estimated from peak to peak time and the viscous damping factor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{\u0026xi;=}\\frac{c}{2m}\\)\u003c/span\u003e\u003c/span\u003e is evaluated by fitting a curve to the experimental data. The \u0026lsquo;c\u0026rsquo; estimated from the experimental data is of order\u0026thinsp;~\u0026thinsp;10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e Ns/m for the impact velocity of 0.5 m/s and it increases by 2\u0026ndash;3 times for an increase in the impact velocity upto 2 m/s. The rise in the damping coefficient with an increase in the velocity is due to the higher rate of energy dissipation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\dot{\\text{\u0026empty;}}\\text{ ~}{\\int }_{\\text{0}}^{{\\text{R}}_{\\text{d}}}\\frac{\\text{2\u0026pi;r\u0026mu;}}{\\text{\u0026delta;}}{\\text{v}}^{\\text{2}}\\text{dr}\\right)\\)\u003c/span\u003e\u003c/span\u003e. Furthermore, \u0026lsquo;c\u0026rsquo; increases slightly (~\u0026thinsp;15%) on softer surfaces. The increase is due to viscoelastic behavior of the softer surface. The softer surface (1:30) deforms about 25 times more as compared to the surface 1:10 which pins the droplet at a larger D\u003csub\u003eeq\u003c/sub\u003e than that on the harder one. Accordingly, the thickness of the layer for dissipation of energy \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\text{\u0026delta;)}\\)\u003c/span\u003e\u003c/span\u003e is smaller as the droplet volume is constant and thus the value of c which is related to energy dissipation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\dot{\\text{\u0026empty;}}\\)\u003c/span\u003e\u003c/span\u003e) is large. The total timescale of oscillation is also slightly reduced (~\u0026thinsp;10%) on soft surface (1:30) as compared to a harder surface (1:10) (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eThe spring constant is estimated as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{k = m}\\left({\\text{\u0026omega;}}^{\\text{2}}\\text{+}{\\left(\\frac{\\text{c}}{\\text{2m}}\\right)}^{\\text{2}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and it signifies the tendency of the system to restore its energy (Chen et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). \u003cem\u003ek\u003c/em\u003e decreases by 75\u0026ndash;80 % for an increasein the impact velocity from 0.5 m/s \u0026ndash; 2 m/s. Since there are no external forces to drive the oscillations, the surface energy which is the only restoration energy of the system would dissipate with time. Since the dissipation is high for higher velocities, the restoration energy is also reduced.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eIn this work, we experimentally investigated the impact of polymer droplets on viscoelastic surfaces. The complex dynamics of impact can be identified as three stages, namely, spreading, retraction and oscillation. The spreading of droplets is mainly governed by the inertia of the droplets. A scaling of the inertia and the surface tension forces gives an estimation of the maximum contact diameter formed during spreading as a function of the Weber number as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\frac{{\\text{D}}_{\\text{max}}}{{\\text{D}}_{\\text{0}}}\\text{ ~}{\\text{ We}}^{\\text{1/4}}\\right)\\)\u003c/span\u003e\u003c/span\u003e. The spreading stage is followed by the retraction of the droplets. The droplet retracts to minimize its surface energy. Softer surfaces restrict the movement of the droplet, thereby reducing the rate of retraction as well. The retraction stage is completely eliminated on softer surface at a higher velocity (2 m/s) due to large deformation which pins the droplet at the maximum contact diameter. The end of the retraction stage is identified by pining of the contact diameter in all the cases. The pinned droplet finally oscillates like a damped oscillator till the equilibrium state. The oscillations dampen at a higher rate at high impact velocity due to high viscous dissipation rate and less energy available for the oscillations. A simple spring-mas-damper system analogy also shows that the damping coefficient which is related to energy dissipation is higher on softer surfaces due to its viscoelastic behavior.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eThe authors declare no financial or personal interests to influence the reported work.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e*Corresponding author\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eBinita Pathak\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eDepartment of Mechanical Engineering\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eIndian Institute of Technology, IIT BHU Varanasi\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eVaranasi-221005, India\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eEmail: [email protected]\u003c/p\u003e\n\u003cp\u003e\u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; \u0026nbsp; [email protected]\u003c/p\u003e\n\u003cp\u003eTel: +91-8105431120\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAlizadeh A, Bahadur V, Shang W, et al (2013) Influence of Substrate Elasticity on Droplet Impact Dynamics. 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J Fluid Mech 517:199\u0026ndash;208. https://doi.org/10.1017/S0022112004000904\u003c/li\u003e\n\u003cli\u003eGart S, Mates JE, Megaridis CM, Jung S (2015) Droplet Impacting a Cantilever: A Leaf-Raindrop System. Phys Rev Appl 3:044019. https://doi.org/10.1103/PhysRevApplied.3.044019\u003c/li\u003e\n\u003cli\u003eHowland CJ, Antkowiak A, Castrej\u0026oacute;n-Pita JR, et al (2016) It\u0026rsquo;s Harder to Splash on Soft Solids. Phys Rev Lett 117:184502. https://doi.org/10.1103/PhysRevLett.117.184502\u003c/li\u003e\n\u003cli\u003eIzbassarov D, Muradoglu M (2016) Effects of viscoelasticity on drop impact and spreading on a solid surface. Phys Rev Fluids 1:023302. https://doi.org/10.1103/PhysRevFluids.1.023302\u003c/li\u003e\n\u003cli\u003eJosserand C, Thoroddsen ST (2016) Drop Impact on a Solid Surface. 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Proceedings of the Royal Society of London 25:498\u0026ndash;503. https://doi.org/10.1098/rspl.1876.0073\u003c/li\u003e\n\u003cli\u003eXu L, Zhang WW, Nagel SR (2005) Drop Splashing on a Dry Smooth Surface. Phys Rev Lett 94:184505. https://doi.org/10.1103/PhysRevLett.94.184505\u003c/li\u003e\n\u003cli\u003eYarin AL (2006) DROP IMPACT DYNAMICS: Splashing, Spreading, Receding, Bouncing\u0026hellip;. Annu Rev Fluid Mech 38:159\u0026ndash;192. https://doi.org/10.1146/annurev.fluid.38.050304.092144\u003c/li\u003e\n\u003cli\u003eZhang C, Zheng Y, Wu Z, et al (2019) Non-wet kingfisher flying in the rain: The water-repellent mechanism of elastic feathers. J Colloid Interface Sci 541:56\u0026ndash;64. https://doi.org/10.1016/j.jcis.2019.01.070\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"experiments-in-fluids","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"exif","sideBox":"Learn more about [Experiments in Fluids](http://link.springer.com/journal/348)","snPcode":"348","submissionUrl":"https://submission.nature.com/new-submission/348/3","title":"Experiments in Fluids","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4480907/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4480907/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDroplet impact on soft surfaces is important in many industrial, biological and agricultural applications. In this paper, we have analysed the dynamics of impact of polymer droplets upon PDMS surfaces. We varied the impact velocity (0.5-2 m/s) and found that impact velocity plays a crucial role in the process. The elasticity of the substrate has also been varied to study its effect upon the droplet dynamics. We delineate the entire process into three different stages and employ force balance equations to identify the governing forces during each stage. The initial spreading is strongly inertia-controlled and the maximum diameter obeys a power-law relation with the Weber number (We.\u003csup\u003e25\u003c/sup\u003e), irrespective of the impact velocity and the surface properties. The viscoelastic nature of the surface has a dominant influence upon the retraction of the droplets. The effect is more prominent at a higher velocity wherein, the droplet retraction is completely eliminated. A damped harmonic oscillator-type analogy shows that the damping is higher on soft surfaces and at higher velocities.\u003c/p\u003e","manuscriptTitle":"Dynamics of Impact of polymer droplets on Viscoelastic Surfaces","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-07 12:24:45","doi":"10.21203/rs.3.rs-4480907/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-08-26T08:01:05+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-06-18T13:38:46+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"233631256094067007982515661580901142100","date":"2024-06-13T03:23:45+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"315911100568097699894174796741013457289","date":"2024-06-13T02:55:42+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"278522961479546509692746496742812344140","date":"2024-06-13T02:50:39+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-06-13T02:35:53+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-05-28T07:12:11+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-05-27T11:52:27+00:00","index":"","fulltext":""},{"type":"submitted","content":"Experiments in Fluids","date":"2024-05-26T17:00:24+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"experiments-in-fluids","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"exif","sideBox":"Learn more about [Experiments in Fluids](http://link.springer.com/journal/348)","snPcode":"348","submissionUrl":"https://submission.nature.com/new-submission/348/3","title":"Experiments in Fluids","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"4a3ab6e7-9f45-40a6-ab0f-a6449c4cdba2","owner":[],"postedDate":"June 7th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-09-30T16:04:24+00:00","versionOfRecord":{"articleIdentity":"rs-4480907","link":"https://doi.org/10.1007/s00348-024-03886-x","journal":{"identity":"experiments-in-fluids","isVorOnly":false,"title":"Experiments in Fluids"},"publishedOn":"2024-09-24 15:57:42","publishedOnDateReadable":"September 24th, 2024"},"versionCreatedAt":"2024-06-07 12:24:45","video":"","vorDoi":"10.1007/s00348-024-03886-x","vorDoiUrl":"https://doi.org/10.1007/s00348-024-03886-x","workflowStages":[]},"version":"v1","identity":"rs-4480907","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4480907","identity":"rs-4480907","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}