Experimental and theoretical study about drop impact on inclined surfaces and substrates with different roughness and wettability

Tesis Doctoral Experimental and theoretical study about drop impact on inclined surfaces and substrates with different roughness and wettability Autor: Paula García Geijo Directores: José Manuel Gordillo Arias de Saavedra Catedrático Guillaume Riboux Profesor Titular Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos Escuela Técnica Superior de Ingeniería Universidad de Sevilla 2023 Tesis Doctoral: Experimental and theoretical study about drop impact on inclined surfaces and substrates with different roughness and wettability Autor: Paula García Geijo Directores: José Manuel Gordillo Arias de Saavedra Guillaume Riboux El tribunal nombrado para juzgar la Tesis arriba indicada, compuesto por los siguientes doctores: Presidente: Vocales: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: A mi familia, fuente de apoyo y amor incondicional A Alber, mi compañero Agradecimientos Finalizando esta tesis no se me pueden venir a la mente más que buenos recuerdos y eso es únicamente debido a la gente que me ha acompañado a lo largo de este camino. Tengo mucho que agradecer y tengo claro por donde empezar: mis directores de tesis. José Manuel Gordillo y Guillaume Riboux son los principales culpables de que esta tesis y mi doctorado sean una realidad. Muchas gracias por confiar en mí y darme esta oportunidad, por todo lo que he aprendido con vosotros, de mecánica de fluidos y de la vida, pero sobre todo por lo bien que me habéis acogido y por hacerme sentir como en casa a pesar de no estar en mi querido León. Transmitís pasión por la ciencia y eso hace que sea un gusto trabajar con vosotros. I also wanted to thank Detlef Lohse for giving me the opportunity to spend a few months at University of Twente. I really enjoyed my stay at Physics of Fluids group where I met a lot of people from the fluid mechanics community doing research in different topics. This experience has definitely helped me to broaden my knowledge, and in general, to broaden my mind. These four months in Enschede would have not been the same without the PoFDocs: Javi, Lorène, Vincent, Alex, Youssef, Udo, Ambre and Pierre. I miss our Monday pubquiz, home made dinners, playing cards and board games and having drinks at our base: Stanislaus. Gracias también a mis amigos de Sevilla y a mis amigas de San Justo - Astorga. Ellos han sido la mejor compañía en la celebración de los buenos momentos, y también han hecho que los malos sean más llevaderos. Pero sin duda, el que más ha aguantado los míticos agobios de Paula, sobre todo estos últimos meses, es Alber. Muchas gracias por estar siempre a mi lado, apoyarme y sacar lo mejor de mí. Por último, aunque para nada lo menos importante: mi familia. Gracias papá, mamá y Nuri. Aunque esta etapa la hayamos tenido que vivir desde la distancia, no me han faltado vuestro cariño y vuestros consejos. Espero que estéis orgullosos de vuestra hija y hermana, porque yo no puedo estar más feliz de la familia que me ha tocado y los valores que me III 1 Introducción What we know is a drop, what we do not know is an ocean. Isaac Newton Algo que nos resulta tan familiar, tan común y cotidiano como el impacto de una gota sobre una superficie ha capturado la atención de numerosos investigadores a lo largo de la historia. Hace ya más de 500 años este fenómeno atrajo la atención de Leonardo da Vinci, quien ilustró en una de las páginas del conocido como Códice Leicester o Códice Hammer [1] las observaciones experimentales mostradas en la figura 1.1 [2]. Leonardo da Vinci comprobó que, tras hacer impactar una gota contra una superficie horizontal, terminan desarrollándose finos ligamentos de líquido que se encuentran distribuidos alrededor de todo el perímetro de la gota y que son eyectados radialmente hacia fuera de la zona de impacto. Más adelante, en 1877, Worthington realizó un análisis detallado de la fenomenología tan variada que surge al impactar una gota contra una superficie horizontal. En su contribu- Figura 1.1 Esbozo de la marca que deja una gota de agua tras impactar en una sustrato horizontal realizado por Leonardo da Vinci en el margen del folio 33r del Códice Leicester [1]. Imagen tomada de la referencia [2]. 1 2Capítulo 1. Introducción Figura 1.2 Ilustraciones de Worthington mostrando sus observaciones experimentales de splash al hacer impactar una gota de mercurio sobre una superficie horizontal. Se identifica la formación de los ligamentos distribuidos a lo largo de la gota desde los cuales se eyectan pequeñas gotas. A tiempos mayores (dibujos de la parte derecha) la gota se ha desintegrado completamente. Imagen tomada de la referencia [3]. ción, Worthington ilustra gráficamente los diferentes patrones que surgían al cambiar las propriedades materiales del líquido y el radio de la gota, así como la velocidad de impacto variando la altura de caída del líquido con respecto a la posición del sustrato [3,4]. La dinámica del impacto de gotas sigue siendo de gran interés en la actualidad, y el motivo principal es que se trata de un fenómeno que está presente no solamente en multitud de situaciones cotidianas, sino también en diversas aplicaciones industriales: impresión con chorro de tinta, fenómenos de transferencia de calor donde es importante la cuantificación del área de contacto líquido-sólido, el estudio de la dispersión y propagación de enfermedades y productos contaminantes, e incluso también en los análisis forenses [5–10]. Como ya anticipó Worthington hace casi 150 años y como se ha ido descubriendo en estudios posteriores, el impacto de una gota sobre una superficie depende de una gran variedad de parámetros. No solo juegan un papel relevante la velocidad de impacto, las propiedades físicas del líquido y el radio de la gota, sino también otras propiedades como la rugosidad y mojabilidad de la superficie, su grado de inclinación, la presión atmosférica y las propriedades físicas del gas ambiente, así como el hecho de que el sustrato esté mojado o completamente seco, entre otros muchos. Esta tesis se centra en el estudio del caso concreto del impacto de gotas milimétricas de líquidos de baja viscosidad a presión atmosférica sobre superficies secas e impermeables a una velocidad de impacto tal que los efectos inerciales dominan sobre los esfuerzos viscosos y las fuerzas de tensión superficial. 1.1 Definiciones previas 3 (a) (b) Figura 1.3 Vista lateral y cenital de los fenómenos de (a) spreading y (b) splashing producidos cuando una gota etanol impacta contra una placa de vidrio horizontal a una velocidad (a) inferior y (b) superior a la velocidad de transición al splash. Imágenes grabadas durante el estudio experimental realizado en esta tesis. Los parámetros que se variarán y, por tanto, las características cuya influencia se analizará, serán la viscosidad del líquido, el tamaño de las rugosidades de la superficie, el ángulo de contacto que forma la gota con el sólido, que está relacionado con el grado de repelencia al agua del sustrato, y el ángulo de inclinación que forma la trayectoria que sigue la gota en el momento del impacto con la pared. 1.1 Definiciones previas Durante los primeros instantes de tiempo tras el impacto de una gota contra una superficie, se pueden producir dos situaciones diferentes. Cuando la velocidad de impacto es baja, la gota se expande suavemente de forma tangencial al sustrato, dando lugar a lo que en la comunidad científica y en la literatura inglesa se conoce como spreading (ver figura 1.3a). Sin embargo, si la misma gota impacta a velocidades cada vez mayores, se observa que existe una velocidad crítica a partir de la cual la gota pierde su integridad, rompiéndose en pequeños fragmentos que son eyectados radialmente hacia fuera a velocidades mayores que la de impacto, fenómeno que se conoce comunmente como splash (ver figura 1.3b) [8,11]. Respecto al fenómeno de desintegración de la gota, algunas contribuciones diferencian entre dos tipos de splash a la hora de analizar el proceso de rotura. Si se eyectan pocas gotas y de forma aproximadamente tangente a la superficie, el proceso se denomina splash puntual o prompt splash, mientras que se emplea el término corona splash para caracterizar aquellas situaciones en las que la lámina de líquido se separa de manera ostensible del sustrato y el número de gotas eyectadas es mucho mayor [12 – 15]. Sin embargo, en esta tesis no se atenderá a dicha distinción y el análisis se simplificará considerando que la 4Capítulo 1. Introducción Figura 1.4 Esquema de las distintas regiones en las que se divide la gota para el estudio del proceso de expansión. Figura 1.5 Imagen experimental de una gota de agua depositada sobre una superficie (a) hidrofílica, (b) hidrófoba. Se comprueba que el ángulo de contacto que forma el líquido con el sólido, que está relacionado con el grado de repelencia al agua, es menor que π/2 para las superficies hidrofílicas y mayor que π/2 para el caso de sustratos hidrófobos. transición de spreading asplashing tiene lugar cuando la gota pierde su integridad y se rompe en gotas más pequeñas y rápidas, que son eyectadas con una velocidad mayor a la de impacto. Durante el proceso de expansión, el campo fluido puede ser dividido en tres regiones bien diferenciadas (ver figura 1.4): la gota, donde el líquido se acelera debido a la presencia de gradientes de presión; la lamella, una delgada lámina de líquido que es eyectada de forma tangente al sustrato desde la región de la gota y donde los gradientes de presión se pueden despreciar debido a su esbeltez; y el rim, el extremo de la lamella que avanza en dirección radial y limita el perímetro de la gota en expansión [16]. Estos procesos de expansión y rotura de la gota son descritos en términos de parámetros adimensionales. Los más comunes, y que serán los empleados a lo largo de esta tesis para caracterizar los distintos fenómenos observados, son los números de Weber, Reynolds y Ohnesorge. El número de Weber mide la importancia relativa entre la presión dinámica y la presión capilar. El número de Reynolds representa la relación entre las fuerzas de inercia y viscosas usando como velocidad característica la velocidad de impacto. Por último, el número de Ohnesorge se define como el inverso del número de Reynolds cuando se toma como velocidad característica la velocidad capilar. En cuanto a la caracterización de la superficie, atendiendo a las propiedades de mojabilidad, se pueden diferenciar dos tipos de sustrato en función del valor del ángulo de 1.2 Experimentos 5 Figura 1.6 Esquema explicativo de la definición del tamaño de grano, ε , de una superficie rugosa, que será el parámetro empleado en esta tesis para caracterizar las irregularidades del sustrato. contacto, θ , que está relacionado con el grado de repelencia al agua. En la figura 1.5 se puede ver una gota del mismo líquido (agua) depositada sobre dos superficies distintas. Los sustratos en los que el líquido forma un ángulo agudo con la pared se denominan hidrofílicos, mientras que si θ>π/2la superficie se considera hidrófoba. Por otra parte, la definición de la rugosidad de un sustrato es muy variada en la literatura. Muchos autores emplean la rugosidad media, Ra , un parámetro muy conocido que representa el valor medio de la altura de los distintos puntos de la superficie respecto al plano medio de elevación del sustrato [13,15, 17,18]. Pero hay otros autores que prefieren usar parámetros menos comunes como el Rpk o altura media de los picos que sobresalen del núcleo de la superficie; la media cuadrática del valor de la ordenada a lo largo de la longitud analizada, Rrms ; o el espesor medio de las irregularidades, Rsm , definido como la media de las longitudes de los picos o valles que conforman la superficie [14,19, 20]. En esta tesis se empleará el tamaño de grano, ε , que se refiere al diámetro medio de las partículas abrasivas que conforman las superficies rugosas de las lijas que se usarán en el estudio experimental, ya que es un valor proporcionado directamente por el fabricante, véase la figura 1.6. De esta manera, las imperfecciones del sustrato quedan definidas con una única longitud, mientras que en otros casos es necesario conocer dos longitudes para determinar el parámetro que define la rugosidad: la altura de los picos, y la separación entre ellos. Además, aunque algunos estudios se han centrado en el análisis del efecto de la rugosidad empleando muestras creadas mediante técnicas de microfabricación donde la distribución espacial y la geometría de las protuberancias están definidas y controladas [21 – 26], esta tesis se enfocará en el análisis de superficies con una distribución arbitraria de las imperfecciones, que son las que se encuentran más habitualmente en las diversas aplicaciones en las que una gota impacta contra un sustrato rugoso. 1.2 Experimentos Con el objeto de comparar y validar los resultados teóricos de los modelos propuestos, en esta tesis se ha utilizado un montaje experimental como el mostrado en la figura 1.7. Una gota con radio R se deja caer desde una aguja con una velocidad V que puede ser modificada 6Capítulo 1. Introducción Figura 1.7 Esquema del montaje experimental empleado para realizar los distintos experimentos reportados a lo largo de esta tesis. variando la altura de caída con respecto a la posición del sustrato sólido. Para ello se bombea un líquido de viscosidad µ y densidad ρ con la ayuda de una jeringuilla de tal modo que la generación de la gota sea cuasiestática. La gota impacta sobre una placa de vidrio cuya superficie puede ser alterada a voluntad, bien aplicando un tratamiento superhidrofóbico o bien solapando lijas de rugosidad controlada y variable. En cualquier caso, estas placas de vidrio, modificadas o no, son siempre sustituidas tras cada experimento. Por último, el ángulo de inclinación χ que forma la superficie de impacto con la horizontal, puede ser variado mediante la ayuda de un goniómetro. Los fenómenos de spreading ysplashing son grabados empleando sendas cámaras de alta velocidad, que graban de manera simultánea los primeros instantes del impacto desde dos vistas diferentes: una lateral y otra cenital. La vista lateral es empleada para determinar el radio y la velocidad de impacto de la gota, mientras que la visión cenital proporciona información cualitativa y cuantitativa de los fenómenos de expansión y rotura de la gota. El posprocesado de las observaciones experimentales se realiza utilizando las herramientas de análisis de imagen incorporadas en el software comercial MATLAB. 1.3 Spreading Cuando la velocidad de impacto es baja, la gota, en lugar de disgregarse, mantiene su integridad y se expande sobre la superficie. Si el impacto es perpendicular al sustrato, el proceso de spreading es axilsimétrico, pero cuando la trayectoria que sigue la gota forma un ángulo con la pared, situación que se da en la mayoría de casos en los que una gota impacta contra una superficie, la expansión del líquido es asimétrica, véase la figura 1.8 [27–31]. En el caso del impacto normal, el proceso de spreading ya ha sido completamente definido en contribuciones previas. Concretamente Gordillo et al. [16] desarrollaron un modelo teórico basado en las ecuaciones de balance de masa y de cantidad de movimiento capaz de describir la física del fenómeno de spreading y de reproducir los resultados 1.3 Spreading 7 Figura 1.8 Proceso de expansión de una gota de agua con glicerina de radio R≃1.25 mm al impactar contra una superficie de acero inoxidable sin tratar, o con un tratamiento hidrofóbico. En las dos primeras filas de imágenes el sustrato se mueve con una velocidad Vs dando lugar a figuras asimétricas, proceso semejante al caso de una gota impactando formando una trayectoria inclinada antes de chocar contra el sólido horizontal y quieto (veáse la figura 1.9). Sin embargo, cuando el sólido está en reposo, en las imágenes de la tercera fila se observa una expansión axilsimétrica con geometrías concéntricas en torno al punto de impacto. Imagen tomada de la referencia [27]. Figura 1.9 Esquema de los distintos tipos de de impacto inclinado abordados en esta tesis así como en otras contribuciones. (a) Impacto de una gota cuya trayectoria forma un ángulo χ con la normal a la superficie. (b) Impacto de una gota sobre una superficie que se traslada paralelamente a sí misma con una velocidad Vsinχ . (c) Impacto vertical sobre una superficie inclinada que forma un ángulo χcon la horizontal. experimentales de la expansión de una gota a lo largo del tiempo en sustratos horizontales de distinta mojabilidad. De entre las investigaciones que han tratado el caso del impacto inclinado de gotas sobre 8Capítulo 1. Introducción superficie sólidas se observan dos modos diferentes de abordar el problema: colocando el sustrato de forma perpendicular a la dirección de la gravedad o no. En el primer caso, se puede dar a su vez dos situaciones: una gota que cae sobre un sustrato horizontal describiendo una trayectoria que no es perpendicular al mismo (ver figura 1.9a), o una gota impactando verticalmente sobre un sustrato horizontal que se encuentra en movimiento con una determinada velocidad tangencial (ver figura 1.9b). Esta última es la idea que se ha utilizado en el montaje experimental de muchas de las contribuciones presentes en la literatura [27 – 31]. Sin embargo, en otros estudios [32 – 35] así como en esta tesis, el impacto inclinado se ha estudiado para el caso de una gota que cae verticalmente sobre un sustrato colocado formando un cierto ángulo sobre la horizontal, fenómeno en el cual la dirección de la gravedad no es perpendicular al sólido, como se muestra en la figura 1.9c. De entre todos ellos, este es el montaje experimental más sencillo a la hora de realizar experimentos de impacto inclinado. Es importante destacar que, como el tiempo característico del fenónemo de spreading es tan corto (de tan solo unos milisegundos), la gravedad no tiene tiempo suficiente para modificar la velocidad del flujo durante el impacto. Además las fuerzas capilares sobre el rim son mucho mayores que las fuerzas gravitatorias. Por lo tanto, se pueden despreciar los efectos gravitacionales en la descripción del proceso de expansión de la gota, y los tres sistemas mostrados en la figura 1.9 pueden ser abordados usando el mismo modelo teórico. A pesar de encontrar en la literatura correlaciones que son capaces de describir las formas asimétricas que se producen durante la expansión de una gota tras el impacto inclinado [27,30,31,35], no existe un estudio capaz de explicar y cuantificar los resultados experimentales partiendo de principios básicos y describiendo la física del problema en cuestión. Por lo tanto, el primer objetivo de esta tesis es intentar rellenar este vacío en el conocimiento aportando tanto nuevos datos experimentales así como una teoría, basada en resultados previos, véase la Ref. [16], que reproduce de manera satisfactoria las observaciones experimentales en lo que concierne a la influencia que posee el ángulo de inclinación de la superficie en la huella que deja la gota tras impactar y que puede ser empleada para describir los tres escenarios mostrados en la figura 1.9. 1.4 Splashing Como se indicó anteriormente, se emplea el término de splash para caracterizar aquellas situaciones en las que una gota, tras impactar contra una superficie, termina disgregándose en otras más pequeñas. Como se puede observar en la figura 1.10, el aire atmosférico juega un papel muy importante en este proceso. De hecho, la influencia del gas de la atmósfera circundante en la aparición de splash fue descrito por Riboux y Gordillo [37, 38] con un modelo teórico capaz de reproducir con precisión tanto los datos experimentales propios como los publicados por otros autores. Riboux y Gordillo explican que, para que tenga lugar la eyección de gotas en el caso de sustratos hidrofílicos lisos y secos, es necesario que el frente de avance de la lamella (el rim) se despegue primero de la pared. Son las fuerzas aerodinámicas sobre esta región las que producen una aceleración vertical y, por tanto, 1.4 Splashing 9 Figura 1.10 Observaciones experimentales del impacto de gotas de etanol sobre placas de vidrio, superficies lisas e hidrofílicas, en distintas condiciones de presión del gas circundante indicadas en cada secuencia de imágenes. Se muestran cuatro instantes de tiempo diferentes: t=0 , 0.276 , 0.552 y 2.484 ms desde el momento del impacto. Las gotas tienen un radio R≃1.7 mm y caen sobre el sólido a la misma velocidad, V≃3.74 ms. Imágenes tomadas de la referencia [36]. inducen una velocidad vertical al rim que hace que éste que se despegue del sustrato. Una vez que esto ocurre, las inestabilidades capilares pueden crecer en la dirección acimutal del borde de la lamella, que termina rompiéndose en gotas que son eyectadas radialmente hacia fuera de la región de impacto. Pero no todas las superficies son lisas e hidrofílicas. De hecho, ninguna superficie puede considerarse perfectamente lisa. Atendiendo al efecto de las imperfecciones del sustrato en el fenómeno de desintegración de la gota, se ha observado experimentalmente que cuanto mayor es la amplitud de las rugosidades, menor es la velocidad crítica de transición al splash, véase la figura 1.11 [12 – 15,17 – 19,39,40]. Sin embargo, hasta la publicación de los resultados contenidos en esta tesis, no existía una teoría basada en principios físicos fundamentales que describiese la influencia que tienen las irregularidades de la superficie en el proceso de rotura de una gota al impactar contra una superficie, ni siquiera en condiciones de presión atmosféricas. Además, existen muchos tipos de superficies que repelen el contacto con el líquido, lo que en la literatura se conoce con el nombre de sustratos hidrofóbos o superhidrofóbos. Este tipo de superficies cada vez son más comunes debido al creciente interés que suscitan por sus diversas aplicaciones prácticas. En efecto, la huella que queda al impactar una gota sobre este tipo de sustratos es menor que sobre una superficie hidrofílica, véase la figura 1.12. Por tanto, las superficies superhidrófobas (es decir, las que repelen el contacto con los líquidos, véase la figura 1.5) se ensucian menos, y tienen mayor capacidad para mantenerse secas y aisladas térmicamente, véase la figura 1.12. Una de las razones que 16 Capítulo 2. Inclined impact of drops tras el impacto. Esta expresión es válida cuando el ángulo de inclinación es bajo. Para mayores valores del ángulo de inclinación, la forma final de la gota se puede aproximar a la de una elipse caracterizada por sus semiejes mayor y menor lo cuales también pueden ser calculados haciendo uso de la teoría propuesta. Las formas finales predichas con el modelo propuesto concuerdan con las observaciones experimentales de las manchas de líquido que permanencen en la pared, exceptuando el hecho de que en la parte inferior de la gota se forma una acumulación de líquido que fluye hacia abajo. Las limitaciones de la teoría para describir la aparición de este chorro líquido también se incluyen en este capítulo. Paper reference: García-Geijo, P., Riboux, G., & Gordillo, J. (2020). Inclined impact of drops. Journal of Fluid Mechanics,897, A12. doi:10.1017/jfm.2020.373 Citations: 13 (WoS 2023) 20 (Google Scholar 2023) 14 (Scopus 2023) FWCI: 1.45 (Scopus 2023) Journal of Fluid Mechanics: Impact Factor: 4.245 (JCR 2021) Category: Physics, Fluids & Plasmas Category Rank: 3/34 (JCR 2021) Category Quartile: Q1 (JCR 2021) Scan the QR code for accessing paper in the publisher website: 3 Spreading and splashing of drops impacting rough substrates Paper reference: García-Geijo, P., Quintero, E., Riboux, G., & Gordillo, J. (2021). Spreading and splashing of drops impacting rough substrates. Journal of Fluid Mechanics,917, A50. doi:10.1017/jfm.2021.313 Open Access Citations: 18 (WoS 2023) 20 (Google Scholar 2023) 17 (Scopus 2023) FWCI: 2.79 (Scopus 2023) Journal of Fluid Mechanics: Impact Factor: 4.245 (JCR 2021) Category: Physics, Fluids & Plasmas Category Rank: 3/34 (JCR 2021) Category Quartile: Q1 (JCR 2021) Scan the QR code for accessing paper in the publisher website: 17 J. Fluid Mech. (2021), vol.917, A50, doi:10.1017/jfm.2021.313 Spreading and splashing of drops impacting rough substrates P. García- Geijo1,E.S.Quintero 1,G.Riboux 1and J. M. Gordillo1,† 1Área de Mecánica de Fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n, 41092 Sevilla, Spain (Received 30 October 2020; revised 24 February 2021; accepted 6 April 2021) Here, we present experimental results of water and ethanol drops of radii R, density ρ and interfacial tension coefficient σ, impacting with a velocity Vover different types of sandpapers containing particles of characteristic diameter εembedded in their surfaces. It is shown that the transition from spreading to splashing at normal atmospheric conditions can be classified depending on the value of the parameter ε/Ht≃Weε=We(ε/R),with We =ρV2R/σ the Weber number and Htindicating the initial thickness of the thin film –the lamella – which is ejected along the substrate once the drop touches the solid. When Weε1 and the liquid wets the substrate, the critical value of the Weber number above which the drop splashes, Wec, can be predicted using the results in Gordillo & Riboux (J. Fluid Mech., vol. 871, 2019, R3) once the angle the advancing rim forms with the substrate, α, is expressed as a decreasing function of the static advancing contact angle. The calculated values of Wecfor the case of water drops impacting over rough substrates are smaller than the corresponding ones for smooth substrates, in agreement with experimental observations. Moreover, if the liquid does not wet the substrate, it is also shown that the splash velocity can be predicted using the theory for superhydrophobic substrates in Quintero, Riboux & Gordillo (J. Fluid Mech., vol. 870, 2019, 175–188). For those cases in which Weε1 and the liquid wets the substrate, we demonstrate that the critical Weber number for splashing decreases with εas Wec∝(Rcos θ0/ε)3/5,withθ0 the value of the Young contact angle. Key words: aerosols/atomization, drops †Email address for correspondence: [email protected] © The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. 917 A50-1 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo 1. Introduction A drop impacting a dry solid substrate will either spread tangentially to the surface without breaking or will splash, disintegrating into tiny droplets ejected radially outwards at velocities far larger than the impacting one (Yarin 2006; Josserand & Thoroddsen 2016). It is now known that the conditions under which splashing occurs does not only depend on the liquid properties, on the impact velocity and on the drop radius, as expressed by the well-known correlation proposed in Mundo, Sommerfeld & Tropea (1995), but also on other parameters such as the surrounding gas pressure (Xu, Zhang & Nagel 2005; Riboux & Gordillo 2014;Stevens2014; Gordillo & Riboux 2019), the roughness (Stow, Hadfield & Ziman 1981; Range & Feuillebois 1998; Xu, Barcos & Nagel 2007;Latkaet al. 2012; Quetzeri-Santiago, Castrejón-Pita & Castrejón-Pita 2019a) or the substrate wettability (de Goede et al. 2018; Quetzeri-Santiago et al. 2019b; Quintero, Riboux & Gordillo 2019), with wetting and roughness intimately related with each other, see e.g. Quéré (2008). It is the purpose of this contribution to analyse the spreading and splashing of droplets of low viscosity liquids such as water and ethanol or mixtures of both impacting at normal atmospheric conditions over rough substrates. These are, precisely, the most common conditions involving the impact of a drop against a solid found in both practical applications and in our daily life experience: indeed think, for instance, of rain drops falling on the sidewalk, which clearly is a rough substrate like the vast majority of solids. Our study will be limited to analysing those cases in which the surface is initially dry, a situation which differs from the similar –albeit simpler case because neither the topography of the substrate nor wetting effects are present – in which the drop falls on a pool or thin liquid film (Josserand & Zaleski 2003; Cimpeanu & Moore 2018). It will also be assumed that the drop falls over the solid perpendicularly because the effects associated with the impact direction (Bird, Tsai & Stone 2009; Almohammadi & Amirfazli 2017; Hao & Green 2017;Haoet al. 2019) can be easily accounted for using the framework put forward in, for instance, Gordillo & Riboux (2019) and García-Geijo, Riboux & Gordillo (2020). As it was mentioned above, Mundo et al. (1995) studied the case of drops impacting at normal atmospheric conditions on either smooth or rough dry surfaces and characterized the spreading–splashing transition through the so-called Kparameter, or splashing parameter, which is nothing but a correlation involving the Reynolds and Ohnesorge numbers based on the liquid properties. Nevertheless, the experiments conducted by Xu et al. (2005)andStevens(2014) revealed that drop splashing can be suppressed by reducing the air pressure and also that the splash threshold velocity behaves non-monotonically for low values of atmospheric pressure, these facts indicating that drop splashing heavily depends on the properties of the surrounding gaseous atmosphere. The correlation found by Mundo et al. (1995), as well as the experimental results found by Xu et al. (2005)and Stevens (2014), were reconciled by the theory presented in Riboux & Gordillo (2014)and Gordillo & Riboux (2019), where it is shown that the splashing of drops is produced by the lift force exerted by the air on the edge of the lamella. The ideas in Riboux & Gordillo (2014) were developed for the case of smooth dry substrates, and it will be one of the purposes in this contribution to check whether they can also be applied to the case of rough solids or not. In their now classical contribution, Stow et al. (1981)analysed the splashing of drops on rough substrates, observing that the critical velocity for splashing decreases with increasing values of the amplitude of the roughness. Stow et al. (1981) also proposed a correlation based on the Reynolds (Re)and Weber (We)numbers and on the surface roughness in order to fit their experimental data and, later on, the experiments reported 917 A50-2 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates in Rioboo, Tropea & Marengo (2001) and Range & Feuillebois (1998) confirmed the observations in Stow et al. (1981) that the critical Weber number for splashing depends on the amplitude of the substrate roughness. More recently, Roisman, Lembach & Tropea (2015) proposed a correlation expressing that the critical Weber number for splashing does not depend on the roughness amplitude but on the slope of the substrate corrugations. In the same vein, but for the case of microstructured surfaces, Xu (2007), Tsai et al. (2010), Lembach et al. (2010), Kim et al. (2014), de Jong, Enríquez & van der Meer (2015)and Yarin, Roisman & Tropea (2017) showed that the transition from spreading to splashing depends on the geometrical arrangement of the micropillars. The influence of both air pressure and surface roughness on drop splashing was analysed experimentally by Xu et al. (2007) and Latka et al. (2012), who found that both aerodynamic forces and the substrate roughness play a role in the splashing of drop, but they did not provide with any type of fit, correlation or theory to quantify their observations. In addition, Josserand et al. (2005) simulated the effect on drop splashing of a single obstacle placed on an otherwise dry and smooth substrate and compared the numerical results with experimental observations. The previous revision reveals that there is a lack of physical understanding of the role played by the surface corrugations in triggering the splash, even at normal atmospheric conditions. Then, based on our own experimental observations, here, we present simplified models which, retaining the underlaying physics, provide with predictions for the splash threshold velocity at normal atmospheric conditions, in good agreement with observations. In addition, it will be shown that the equations describing the spreading of drops deduced in Gordillo, Riboux & Quintero (2019) can also be used to predict the observations with rough substrates. While Rioboo et al. (2001), Xu et al. (2007), Latka et al. (2012) and Hao (2017) establish a difference between two types of splashing namely, prompt splashing and corona splashing, here, we will make no distinction between them and will simply determine the conditions for which the drops keep their integrity after the impact (spreading) or they break into smaller and faster droplets (splashing). Let us point out here that this study focuses on the most common case of applications in which neither the spatial distribution nor the geometry of the protuberances of the rough substrate are controlled using microfabrication techniques. The paper is structured as follows: in §2we describe the set-up and present the experimental results, § 3is devoted to providing the theoretical models for the spreading and the splashing of drops and to showing comparisons of the predictions with the experimental observations. The main conclusions are presented in § 4. 2. Experimental set-up and phenomenology Figure 1 is a sketch of the experimental set-up used to produce water or ethanol drops of radii Rimpacting over different types of sandpapers, these being replaced after each measurement. The drops produced in this way fall with a variable and controllable velocity Vwithin the range of values indicated in table 1. The side and top views of the drop impact process are extracted from the analysis of the videos recorded using two different high-speed cameras: figure 1 shows that a Phantom V710, operated at 33 000 f.p.s. (frames per second) is used to get the images from above with a spatial resolution of 42 μmper pixel, whereas a Phantom V7.3, operated between 11 000 and 13000 f.p.s. is employed to get the lateral views with a spatial resolution of ∼18 μmpixel −1. With the purpose of analysing the effect of wettability on the spreading to splashing transition, two different types of substrates have been employed: here, we make use of high quality silicon-carbide sandpapers with either paper or cloth backing and of aluminium-oxide lapping films 917 A50-3 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo (e) (k)(l) (d) 60° (i) gz y V x (j) (A) (B) (g) (h) (a) (b) (c) (f) Figure 1. Sketch of the experimental set-up used to record the side views of the falling drops (A) and also the views from the top of the spreading or splashing processes (B). The set-up consists of a glass slide Knittel 76 ×26 mm (a) to which a portion of sandpaper is fixed thanks to double-sided scotch film. The sample is removed and replaced after each experiment. The glass slide is placed at a constant height over a vertical travel L200 Thorlabs Lab Jack (b) which is rigidly fixed to a non-vibrating table Photon Control (c). Drops are produced quasi-statically injecting either water or ethanol through a Biolin Scientific C209-22 metallic needle of outer diameter ∼0.7 mm (d) by means of a 1 ml threaded plunger glass syringe Hamilton model 81 441 (e). The height of the injection needle and, hence, the drop impact velocity V, can be changed by means of Standard Thorlabs rails ( f). Experimental images are recorded using two different high-speed cameras. A Phantom camera V7.3, which is not shown in the sketch, is coupled with an objective Edmund Industrial Optics 4×pointing in the direction of a light source Schott KL2500 LCD (g), aligned with the xaxis of the sketch; a light diffuser Thorlabs DG100x100 (h) is used to improve the quality of the images obtained. The images recordedinthiswayareofthetypeshownin(A). A second camera Phantom V710 with a Sigma 105 mm DG Macro objective (i) records the type of images depicted in (B). It is mounted in a tripod Manfrotto Model #028 (j), used to either rotate the camera around the xdirection or to move the camera along the zaxis in the sketch. The images recorded using this second camera are illuminated thanks to a second light source, emitted by a Schott KL2500 LCD (k) which passes through a Thorlabs diffuser DG100x100 (l), placed in the plane (x–z). with polyester backing, hereinafter denoted as SC and AO sandpapers, respectively. The substrate roughness will be characterized by means of the so-called grit size ε, which refers to the average diameter of the abrasive particles embedded in the surface, as shown in figure 2(a), because this value is provided directly by the manufacturers. The values of the grit sizes for SC sandpapers are usually characterized by the European standard FEPA (Federation of European Producers of Abrasives), which is identical to the ISO (International Organization for Standardization) standard, while those for AO sandpapers are indicated through its colour, see table 1. The measurement of the substrate roughness through the grit size εdiffers from that used by Range & Feuillebois (1998), Xu et al. (2007), Tang et al. (2017) and Hao (2017), who employed Ra,seefigure 2(a), or by Latka et al. (2012), Quetzeri-Santiago et al. (2019a) and Roisman et al. (2015), who made use of other parameters such as Rpk or the root mean square roughness, Rrms. The relationship between Raand the grit size εhas been determined here by performing measurements using samples of all types of AO sandpapers listed in table 1 as well as the finest SC substrates. In these type of experiments, where εis known, different images of the substrate topography have been recorded using a laser scanning confocal microscope, see figure 2(b). The images obtained in this way have been later on analysed by means of 917 A50-4 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates Sandpaper εRafS dq We θθ rec θadvθ0 (μm) (μm) Water Ethanol SC P220 68 10.07 ∼3 — 43–265 64–376 32 ±5◦21 ◦54 ◦73 ◦ ———— — — 56±6◦49 ◦69 ◦79 ◦ SC P500 30 4.97 ∼3 — 42–256 64–382 65 ±3◦54 ◦76 ◦81 ◦ SC P1000 18 3.36 ∼3 — 43–256 64–391 74 ±1◦61 ◦76 ◦84 ◦ SC P2400 9 2.15 2.40 5.30 36–242 68–377 31 ±4◦27 ◦44◦69 ◦ SC P4000 5 1.61 2.61 5.22 37–223 64–370 98 ±2◦75 ◦122 ◦SH SC P6000 4 1.48 2.66 6.19 40–225 — 97 ±4◦71 ◦124 ◦SH SC P8000 3 1.35 2.66 5.17 39–225 — 100 ±6◦80 ◦135 ◦SH AO Green 30 4.97 2.77 8.83 38–269 — 41 ±2◦7◦40 ◦74 ◦ AO Blue 9 2.15 2.26 4.77 37–269 — 28 ±3◦10 ◦41 ◦67 ◦ AO Pink 3 1.35 2.03 3.81 37–267 — 43 ±2◦8◦42 ◦69 ◦ AO Lime 1 1.08 1.06 0.40 35–267 — 70 ±2◦66 ◦95 ◦71 ◦ AO White 0.3 0.30 1.08 0.46 34–265 — 90 ±4◦76 ◦102 ◦90.0◦ Table 1. In this experimental study, two different types of liquids, water and ethanol, have been employed. The radii of the drops do not vary much for each of the two liquids used and, consequently, in this contribution we will take the constant values R=1.44 mm for water drops and R=1.05 mm for ethanol drops. Using these values of Rand of the standard material properties for water and ethanol at 25 ◦C, the values of the Ohnesorge number defined in (2.4a–d) are constant for each of the two liquids and equal to Oh =3.1×10−3for the case of water and Oh =7.3×10−3for the case of ethanol. However, the value of the Weber number, We,defined in (2.4a–d) is varied within the ranges indicated in the table. The roughnesses of the two types of sandpapers employed here are classified using the grit size, following the FEPA/ISO designation for SC sandpapers and the surface colour for AO sandpapers. The given values of εand those of Racalculated using (2.1)areprovided in the table, as well as the values of the roughness ratio fand of the dimensionless parameter measuring the slope of the corrugations, Sdq,definedin(2.2). For the case of water drops and for each of the two types of sandpapers used, the table includes the values of the macroscopic static contact angle θ, which is measured from high resolution images taken at the scale of the drop, which is much larger than the grit size. For the case of water drops, the values of the Young angle θ0, of the advancing θadvand of the receding θrec contact angles are also provided in the table. The values of θ0are calculated using the measured values of θ, as illustrated in figure 3. The values of θadvand θrec are measured from high resolution images taken at the instant when drops placed over inclined substrates start sliding at the critical inclination angle (Extrand & Kumagai 1995). For the case of ethanol drops, θ≃θadv≃θrec ≃0. For the case of the SC P220 our experiments reveal that, indeed, there exists two different possible values of the macroscopic contact angle θwhich, consistently, correspond to almost the same value of the Young angle θ0. the software SensoMAP Premium 7.4.8114 with the purpose of calculating the value of Ra=1/AA|z(x,y)|dxdy,withAthe sampling area. A least-square fitting reveals that Raand εare related with each other through equations Ra=0.943 +0.134εfor Ra≥1μm, Ra=εfor Ra<1μm,(2.1) with both Raand εexpressed in μm. Equation (2.1) will be used in what follows to express Raas a function of εand vice versa. The topography of the rough substrate will also be characterized in terms of the parameter measuring the slope of the corrugations, also given in table 1, Sdq =    1 AA∂z(x,y) ∂x2 +∂z(x,y) ∂y2dxdy,(2.2) 917 A50-5 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo Adhesive (a) (b)(c) Abrasive grains Grit size Backing Surface topography Mean plane z x Sample length, l 250 µm µm µm µm µm µm 200 150 100 50 00 100 200 300 25 15 10 5 0 20 050 100150 200 250 50 100 150 15 10 5 0 NM Ra = 1 – A ∫∫A|z(x, y)| dx dy E Figure 2. (a) Sketch showing two of the different parameters used in this study in order to characterize the surface roughness: the grit size, ε, and the arithmetical mean roughness, Ra. The grit size is provided by the manufacturer since it represents the characteristic diameter of the particles used in the fabrication process, whereas Radepends on the surface topography. The value of Rais determined using profilometers or microscopes and it is defined as the average value of the height of each point with respect to the arithmetical mean of the surface elevation throughout the sample length (Gadelmawla et al. 2002). The peaks, placed above the mean plane, are represented in dark grey, whereas the absolute values of the ordinates of the points located below the mean plane, are depicted in blue. (b)Two-andthree-dimensional views of the surface topography corresponding to a sample of P2400 SC sandpaper – see table 1 – obtained by means of the software SensoMAP Premium 7.4.8114 from the analysis of the images obtained with a laser scanning confocal microscope. where zindicates the elevation of the peaks. Quéré (2008) analysed the set of experiments by Onda et al. (1996)and Shibuichi et al. (1996)andprovided useful equations relating θ, which is the contact angle the liquid forms with a rough surface, with the Young angle θ0, defined as the static angle the liquid forms with a flat solid made of the same material. For the case the drop rests on the substrate in the Wenzel regime, which is the case for water drops on the sandpapers in table 1 with larger values of ε, the relationship between θand θ0can be expressed as, see Quéré (2008), cos θ=fcos θ0,(2.3) with fthe roughness factor given in table 1 and in figure 3(a), defined as the ratio between the areas of the rough and the flat surfaces. Figure 3(a)showsthatfincreases with Ra(ε), reaching a plateau for rougher surfaces. Table 1 also reveals that the SC sandpapers with ε<5μm, which are fabricated using a cloth backing instead of the paper backing used in the other types of SC sandpapers considered here, possess a superhydrophobic-like (SH) behaviour. 917 A50-6 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates 3 2 1 f 0 1.0 (b) (a) [1] Superhydrophobic [1] Wenzel [1] Superhydrophilic f = 2.5 f = 3.5 0.5 0 –0.5 –1.0 –1.0 –0.5 0 cos θ0 θ = 26 ° θ = 32 ° θ = 56 ° θ = 65 ° θ = 74 ° cos θ 0.5 1.0 0246 Ra (µm) 810 Figure 3. (a) Value of the roughness factor fas a function of the measured mean roughness, Ra,forthe different types of sandpapers in table 1 with Ra<10 μm. (b) Figure adapted from Quéré (2008): the relationship between the cosine of the macroscopic contact angle, θ, and the Young angle, θ0, can be calculated using the results in [1] of Quéré (2008), obtained analysing the experimental data in Onda et al. (1996)and Shibuichi et al. (1996), represented with open circles. This figure includes the curves corresponding to the case in which the drop rests on the substrate in the Wenzel regime for two different values of f, whereas the horizontal lines illustrate how θ0is calculated using the measured values of θ. In the remainder of this contribution, lengths, velocities, times and pressures will be made dimensionless using R,V,R/Vand ρV2as the characteristic values of length, velocity, time and pressure. Therefore, the drop spreading and splashing processes at normal atmospheric conditions will be characterized in terms of the following dimensionless parameters: We =ρV2R σ,Oh =μ √ρRσ,Re =√We Oh and =ε R,(2.4a–d) with ρ,μand σindicating the liquid density, viscosity and interfacial tension coefficient, respectively. The experimental values of We,and Oh explored, as well as the values of the macroscopic static contact angle θ, are provided in table 1, where it is also shown that the grit size varies between ε∼0.3andε∼68 μm which results, using (2.1), in values of the mean roughness varying between Ra∼0.30 and Ra∼10.1μm. Let us point out here that, since ε∼0.014 μm for the case of smooth glass slides (Hao 2017; Quetzeri-Santiago et al. 2019a), in the following, the value ∼10−5will be used to characterize the experiments corresponding to millimetric water or ethanol drops falling over smooth substrates. As was already pointed out in the introduction, no distinction will be made between prompt and 917 A50-7 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo (a) (b) WeE ≤ 1WeE > 1 Ht VtE (c) Figure 4. (a) Sketch showing the definitions of the initial thickness of the lamella, Ht, as well as the initial velocity of the rim, Vt. The type of splash crucially depends on whether Weε<1 namely, Ht>ε(b)orWeε>1 namely, Ht<ε(c), see (2.6). corona splashing and here we will focus on determining the conditions under which the drops keep their integrity and spread or they break and splash, ejecting faster droplets. Usually, whenever the impact velocity is slightly larger than the splash velocity, many tiny droplets can be depicted in the experimental images (Riboux & Gordillo 2015), with the total volume of the liquid ejected increasing with V,as described in Burzynski, Roisman & Bansmer (2020), who quantified their observations in terms of the parameter βdefined in Riboux & Gordillo (2014). Once the experimental set-up has been described and the surface roughness has been characterized in terms of the grit size ε, the rest of the section is dedicated to present the rich phenomenology arising after the impact of a drop on a rough substrate. The analysis of the different experimental information presented next makes use of previous results in Riboux & Gordillo (2014), where it is found that the rim thickness and velocity at the instant the lamella is initially ejected, Htand Vtrespectively – see figure 4 – can be expressed, in the limit of low values of the Ohnesorge number of interest here, in terms of the drop radius R, the impact velocity Vand the Weber number defined in (2.4a–d)as Ht≃RWe−1,Vt≃(√3/2)VWe 1/3.(2.5a,b) Equations (2.5a,b) have been deduced taking into account that Vt=(√3/2)Vt −1/2 e, Ht=R(√12/π)t3/2 eand also that, in the limit Oh 1, te=1.05 We−2/3,withtethe dimensionless instant the lamella is initially ejected. It will be shown next that the ratio between the grit size εand the thickness of the lamella Htwhich, making use of the (2.5a,b), can be expressed as Weε=ε Ht=ρV2ε σ,(2.6) 917 A50-8 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates t = 0 t = 1 3 mm t = 2 t = 3 t = 4 t = 5 t = 6 = 2.1 × 10–3 = 2.8 × 10–3 = 3.4 × 10–3 = 6.9 × 10–3 Figure 13. Spreading of water drops impacting over SC sandpapers for We =65 ±5, a value below the critical Weber number for splashing, for the values of indicated in each of the rows. The rim limiting the expanding liquid sheet first reaches a maximum and then retracts. The blue line represents the solution of the system (3.1)intheSH limit, with γ=1andβ=1. Equations (3.4a,b) are used to describe the rim retraction process once the drop reaches the maximum spreading radius. The horizontal line serves to indicate the location of the impact point. Gordillo et al. (2019) can be used to predict the spreading of drops on rough substrates under those experimental conditions for which Weε1. 3.1. Spreading of drops for Weε1 For drop impact velocities below those producing the splash transition, the time evolution of the rim position and thickness will be described here using the theory in Gordillo et al. (2019), where t=0 indicates the instant the drop first contacts the solid at the so-called impact point, which is also the origin of radial distance r=0. In Gordillo et al. (2019), the flow is divided into the following three different spatial regions, sketched in figure 14: (i) The drop region, 0 ≤r≤√3t, where the liquid is accelerated by pressure gradients, with r=√3tindicating the radius of the circular wetted area (Riboux & Gordillo 2014). (ii) The lamella, which extends along the spatio-temporal region √3t≤r≤s(t),is located in between the impacting drop and the rim. Since the lamella is slender, pressure gradients can be neglected. (iii) The rim, which is located at a r=s(t), possesses a thickness b(t)and moves with a velocity v(t). The rim refers to the edge of the expanding liquid film limiting the perimeter of the spreading drop. The differential equations describing the time evolutions of s(t),b(t)and v(t)are deduced from the balances of mass and momentum applied at the rim γπ 4 db2 dt=[u(s,t)−v]h(s,t), ds dt=v, γπb2 4 dv dt=[u(s,t)−v]2h(s,t)−(1+β)We−1, ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (3.1) 917 A50-15 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo Drop Lamella b(t) v(t) Rim α h(r, t) u(r, t) r s(t) 3t √— Figure 14. Sketch showing the different variables used in § 3. The flow is divided into three different parts: the drop, where pressure gradients cannot be neglected and which extends along the spatio-temporal region 0 ≤ r≤√3t; the lamella, where pressure gradients can be neglected and which extends along the spatio-temporal region √3t≤r≤s(t), and the rim. The rim radial position, thickness and velocity are indicated, respectively, by s(t),b(t)and v(t)whereas u(r,t)stands for the averaged radial velocity within the lamella and h(r,t) represents its thickness. with u(r,t)and h(r,t)indicating, respectively, the averaged radial velocity and the thickness of the lamella extending along the spatio-temporal region √3t≤r≤s(t)(see figure 14). The values of the parameters γand βin (3.1) are chosen assuming that the shape of the rim is either a circle or semicircle of diameter band also depend on whether the rim wets or not the substrate. In the latter case, which corresponds to a SH-like behaviour, the rim is a circle and hence γ=1andβ=1 whereas, in the former, corresponding to a hydrophilic-like behaviour, the rim is assumed to be a semicircle and then, γ=1/2and β=0. The ordinary differential equations for s(t),b(t)and v(t)are solved particularizing at r=s(t)the following analytical expressions for uand hdeduced in Gordillo et al. (2019), valid for Re 1: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(r,t)=r t−Re−1/2 t√3χx 2ha(x)+2√3λ 7ha(x)x5/2(t7/2−x7/2)+O(Re−1), h(r,t)=9t2 r4ha[3(t/r)2]+Re−1/2 rt √3 2x2+√3(105 χ−60λ) 42 x3(t−1−x−1) +24√3λ 105 x−1/2(t5/2−x5/2)+O(Re−1), (3.2) with λ=1, χ=2/3, x=3(t/r)2and ha(x)indicating the polynomial function also given in Gordillo et al. (2019). The ordinary differential equations in (3.1) are solved subjected to the following initial conditions, imposed at the dimensionless instant tethe lamella is initially ejected (Riboux & Gordillo 2014): At t=te=1.05 We−2/3,s(te)=√3te, v(te)=√3/te 2and b(te)=√12 π t3/2 e.⎫ ⎪ ⎬ ⎪ ⎭ (3.3) 917 A50-16 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates t = 0 t = 1 3 mm t = 2 t = 3 t = 4 t = 5 t = 6  = 0.2 × 10–3  = 0.7 × 10–3  = 20.0 × 10–3 Figure 15. Spreading of water drops impacting over AO sandpapers for We =96 ±4, a value below the critical Weber number for splashing, for the values of indicated in each of the rows. The blue line represents the solution of the system (3.1)forγ=1andβ=1. Equations (3.4a,b) are used to describe the rim retraction process once the drop reaches the maximum spreading radius, but only for =0.2×10−3and =20.0× 10−3because the rim pins to the substrate once the maximum radius is reached for the case =0.7×10−3. The horizontal line serves to indicate the location of the impact point. It is explained in García-Geijo et al. (2020) that the system of ordinary differential equations (3.1) is integrated from the ejection time t=teup to the instant t∗for which the rim velocity vanishes namely, v(t=t∗)=0, with t∗calculated solving the system (3.1). Thereafter, there exist two different possibilities depending on whether the rim wets or not the substrate: for the hydrophilic-like case, the rim pins to the solid and, thus, s(t>t∗)=s(t=t∗)but, for the SH-like case, the rim retracts, namely v(t>t∗)<0. The rim contraction process is described using the results in García-Geijo et al. (2020), where the differential equations in (3.1) are simplified by neglecting the relative fluxes of mass, (u−v)h, and momentum, (u−v)2h,giving the following analytical expressions for s(t>t∗)and v(t>t∗): v=−8 πb∗2We (t−t∗)and s=s∗−4 πb∗2We (t−t∗)2,(3.4a,b) with s∗and b∗the values of the rim position and thickness at the instant t=t∗calculated by integrating the system (3.1). The comparison between the predictions and observations in figures 12,13 and 15 for the two types of liquids and for the two different types of rough substrates (AO and SC) considered in this study, validate the approach presented here for arbitrary values of and t10 whenever Weε1. Indeed, it can be appreciated from figures 5–7that the drop disintegrates right after touching the substrate for Weε1and so the description for the spreading process provided here cannot be applied when the grit size is larger than the thickness of the lamella. Let us point out here that the spreading of the drop along the substrate caused by capillarity, taking place at time scales tO(10),is not the subject of this study. 3.2. Splashing models for Weε1 Since the splashing criterion differs depending on the wetting properties of the substrate, here we consider the following two cases. 917 A50-17 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo 3.2.1. Splashing model for hydrophilic-like behaviour It was explained in Riboux & Gordillo (2014) that the splashing of drops impacting partially wetting substrates takes place when the vertical velocity imparted to the edge of the expanding sheet is larger than the radial growth of the rim, which is caused by capillary retraction. This condition can be written as (Gordillo & Riboux 2019) Klμg μOh We5/6≃0.034,(3.5) with μgthe gas viscosity, We and Oh defined in (2.4a–d)andKla coefficient that accounts for the effect of the gas lubrication pressure in the wedge region formed between the substrate and the advancing liquid front, see figure 14. The value of Klfor normal atmospheric conditions is calculated using the expression given in Gordillo & Riboux (2019) Kl=3 tan2αln Aμ μg3/4 Oh−1/4We λg R−1,(3.6) with λg=68 ×10−9m the mean free path of gas molecules, A=0.011 a constant and α the wedge angle sketched in figure 14, whose slight variations around α∼60 ◦, already pointed out in Gordillo & Riboux (2019), will be later on expressed as a function of the static advancing contact angle, θadv,seetable 1. 3.2.2. Splashing model for hydrophobic-like behaviour It was explained in Quintero et al. (2019) that the splash transition for the case of SH substrates takes places when the time characterizing the radial growth of the rim, Th=(R/V)th=(R/V)(1/bdb/dt)−1, is substantially larger than the capillary time, Tc= (R/V)tc=(ρR3b3/8σ)1/2. Indeed, for capillary corrugations to be amplified up to the instant the drops are ejected from the rim, it is necessary that tc/th0.1 (Riboux & Gordillo 2015), which yields the following splash criterion: tc th=b/8db dtWe1/20.1,(3.7) with b(t)the thickness of the rim (see figure 14) calculated through the integration of (3.1). Hence, capillary instabilities will only break the rim for values of the Weber number above a certain threshold, Wec,givenby(3.7). Figure 16 illustrates how the value of Wec is determined from the solution of (3.1), (3.2)and(3.7)forλ=1. Let us point out here that the results in figure 16 reveal that the values of the critical Weber numbers differ from those calculated in Quintero et al. (2019): indeed, we found a small typo in the code used to solve (3.7), which has been corrected here and which does not affect any other of the results presented in our earlier work. In fact, we have verified that the experimental values for the splash threshold velocity corresponding to SH substrates measured in Quintero et al. (2019) are very well predicted by the solution of (3.7) with smaller values of λ (λ<1) in (3.1)and(3.2), a fact indicating that, as expected, the more slippery is the non-wetting substrate, the smaller is the value of the friction factor λ. Notice that (3.7) describes the spreading to splashing transition in cases of drops with Oh 1 impacting over non-wetting dry substrates at normal atmospheric conditions. Indeed, the gas pockets entrapped at the corrugations, which are responsible for the SH-like behaviour depicted in figures 6 and 7, could not be present for small values of the ambient pressure. 917 A50-18 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates 0.30 We = 60 We = 90 We = 127 We = 180 tc/th = 0.1 (a)(b) 0.25 0.20 0.15 tc/th 0.10 0.050 0.2 0.4 t 0.6 0.8 1.0 250 200 150 100 Wec 50 2.0 2.5 3.0 Oh 3.5 4.0 (×10–3) 4.5 5.0 Figure 16. (a) Time evolution of the ratio tc/thdefined in (3.7) calculated solving (3.1)and(3.2)forγ=1, β=1, λ=1anddifferent values of the Weber number for Oh =3.1×10−3. Notice that the ratio tc/this below the threshold value 0.1 only for values of the Weber number We 125. (b) Dependence of the value of the critical Weber number, Wecon Oh for λ=1. 101 100 Vt V 102 ~ 0 2.10 3.50 12.5 20.8 0.20  3/2We1/3 0.70 2.10 6.30 20.8 We Figure 17. The measured velocity at which the lamella is initially ejected is in good agreement with the predicted value, Vt/V≃√3/2We1/3, given in (2.5a,b), for the AO and SC substrates in table 1.Thevalues accompanying each of the symbols in the legend – a triangle for glass, circles for SC sandpapers and squares for AO sandpapers – indicate 103×. 3.3. Splashing criterion for wetting substrates and Weε1 The experimental evidence shown in figures 5–7reveals that, for the cases in which the grit size is such that ε>Htor, equivalently Weε1, a lamella is not formed and the drop disintegrates producing roughly cylindrical fingers which break as a consequence of the growth of capillary instabilities. Then, for the drop to splash in this regime, it is first necessary that a jet with a typical diameter ∼εis formed, which implies that the liquid velocity Vt=(√3/2)VWe 1/3at the instant when it is ejected along the substrate protuberances, is larger than the Taylor–Culick velocity VTC ∝σ/(ρε), (3.8) with the exact prefactor in (3.8) given in e.g. Hoepffner & Paré (2013). Figure 17 shows that, indeed, the lamella is ejected at the velocity predicted by (2.5a,b) and, in spite of 917 A50-19 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo (a) (c)(b) Vw Vt <Vw Vt Vt >Vw θ0 εε ε ε ε ∼ Vt Figure 18. (a) Sketch illustrating one of the main ideas behind the model developed to predict the splash transition for Weε1: we assume that the liquid flows along channels of characteristic width εand that splashing occurs when the velocity Vt–seefigure 17 – of a jet with a characteristic thickness ∼εis larger than both the Taylor–Culick velocity (3.8) and the wetting velocity Vwgiven in (3.9). Here, both the Taylor–Culick velocity and the wetting velocity given in (3.9) are calculated with the simplifying hypothesis that the liquid flows along cylinders with a diameter ε,andso similar ideas to those in Quéré (1997) can be used to calculate Vw.Panel (b) illustrates that no splash will occur if the liquid velocity Vt–seefigure 17 –issmallerthanthe wetting velocity, Vt<Vw.Panel (c) illustrates a case in which splash occurs because the liquid velocity Vtis larger than the wetting velocity, Vt>Vw. the fact that we could not record experimental images with sufficiently good spatial and temporal resolutions so as to measure the dependence of the thickness of the ligaments with the substrate roughness, our qualitative measurements in the Appendix and the results in Xu et al. (2007) indicate that the width of the ejected fingers depends on ε.However, (3.8) is not the only restriction for a ligament to be formed: it is also necessary that Vt is faster than the velocity Vwwith which the liquid wets the interstices formed over the rough substrate, see the sketch in figure 18(a). Indeed, in this way, the height of the liquid film along the ’channels’ of width εincreases in time until it is larger than the grit size ε, producing the ejection of a liquid thread above the corrugated substrate, which acts as a ramp if, in addition, Vt>VTC. Since, for the low viscosity liquids considered here, the Ohnesorge number based on εis such that Ohε=Oh−1/21forWeε1, the wetting velocity Vwcan be calculated assuming the simplifying hypothesis that the liquid flows along a cylinder of diameter ε–seefigure 18 –as 1 2ρV2 w∼4σcos θ0 ε⇒Vw∼8cosθ0VTC,(3.9) with VTC given in (3.8)andθ0the Young contact angle, which slightly differs from the experimental value of θ,see(2.3)andtable 1. Notice that the balance expressed by (3.9) 917 A50-20 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates between the dynamic pressure and the capillary pressure beneath a spherical meniscus forming an angle θ0with the walls of a circular channel of diameter ε, is proportional to that reported by Quéré (1997) in his analysis of the initial instants of the capillary rise of low viscosity liquids in cylindrical tubes. It is expected that (3.9) holds if √νTc ε=√νε/Vt ε=Re−1/2We−1/6−1/21⇒We−5/12Oh1/2−1/21,(3.10) namely, if the width of the boundary layer developed during the characteristic residence time Tc∼ε/Vt=ε/VWe−1/3is much smaller than the width of the channel ∼ε,a condition which is clearly verified here because We 1andOh 1. Notice that the same type of balance as that expressed by (3.9) between inertia and capillary pressure, holds during coalescence of two drops of radii R(Biance, Clanet & Quéré 2004; Winkels et al. 2012) or when a drop wets a wall, a process which was found by Bird, Mandre & Stone (2008) to be influenced by the value of the static contact angle but not by the viscous dissipation at the advancing contact line. In view of the discussion above, it is expected that the drop will splash when the most restrictive of the conditions Vt VTC 1⇒We −3/5and Vt Vw 1⇒We Kw(8cosθ0)3/5−3/5,(3.11a,b) with Kwan order-unity constant to be determined from experiments, is satisfied. The reason for the constant Kwin (3.11a,b)isthat(3.9) rests on the assumption that the geometry of the rough substrate can be viewed as that of a cylinder of diameter ε.Itis expected that the clear differences existing between the real geometry and that of a cylinder can be accounted for through the adjustable constant Kw. The values of Weccalculated using the theoretical approximations explained above are compared with our own experimental data in figure 19, whereas the comparisons between experiments and the correlations proposed by Range & Feuillebois (1998) and Tang et al. (2017), are provided in the Appendix. The experimental data in Range & Feuillebois (1998), corresponding to values of Oh ≃3×10−3, are also included in figure 19.Sincethe prediction in (3.11a,b) does not depend on Oh, those experimental data for which Weε1 in Hao (2017), Tang et al. (2017), as well as those corresponding to the normal impact of drops on rough substrates in Aboud et al. (2020), have also been included in figure 19. Notice that our description is limited to <0.1 since, for larger values of the substrate roughness the separation of scales between the grit size εand the drop radius Ris small and the splash transition will depend on the local geometry of the substrate around the impact point. For those cases in which the rim wets the substrate and Weε1, the values of the critical Weber number for splashing have been calculated using the results in Gordillo & Riboux (2019), where it was pointed out that the wedge angle α≈60 ◦in (3.5) slightly depends on the wetting properties of the surface, with larger values of αfor smaller values of the contact angle. This fact justifies the following empirical equation for α: α=62.5◦−9◦[θadv/90 ◦],(3.12) where the value of α(θadv=0)=62.5◦≈60 ◦corresponding to liquids such as ethanol, methanol or acetone was already reported in Riboux & Gordillo (2014)andthesmall prefactor 9◦/90 ◦=0.1 has been chosen in order to maximize the agreement between experimental measurements and predictions. 917 A50-21 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo 10–3 10–6 10–5 10–4 10–3 10–2 10–1 (1) Water (2) Water (2) Ethanol (3) Water Present study water Eq. (3.5) Water, α = 55 ° Eq. (3.5) Water, α = 58 ° Eq. (3.5) Ethanol, α = 62.5 ° Eq. (3.7) λ = 1 Present study Ethanol Present study WE05% Present study WE20% Present study WE60% (4) Water (3) Ethanol 10–3 10–2 10–1 100 10–4 10–2 102102 102 Wec Wec Weε = 1 Wec = 5 (8 cos(0 °))3/5 –3/5 Wec = 5 (8 cos(75 °))3/5 –3/5 Wec = 5 (8 cos(68 °))3/5 –3/5 Wec = 10 –3/5 Wec = 5 83/5( /cos θ0)–3/5 /cos θ0 10–1 (b) (a) (c) Figure 19. (a) For the case of wetting substrates and Weε1, the values of the critical Weber number have been calculated using (3.5)and(3.12)withOh =7.3×10−3(ethanol) and Oh =3.1×10−3(water) whereas, for the case of water drops and non-wetting substrates, the value of the critical Weber number has been calculated using (3.7)withtc/th=0.1, λ=1andOh =3.1×10−3. The values of Wecfor Weε1have been calculated using the second equation in (3.11a,b) with the value of the free constant adjusted to Kw=5 and using the values of the Young contact angle θ0given in table 2. The experimental data points in Range & Feuillebois (1998), Tang et al. (2017) and Aboud, Wood & Kietzig (2020) have been included in the figure once (2.1) has been used to express Raas a function of ε. Here, [1] Range & Feuillebois (1998), [2] Hao (2017) and [3] Tang et al. (2017); the experimental data in [4] Aboud et al. (2020)forWeε0.5 are also included here because figures 6 and 7show that the rough-induced transition to splashing for the case of water takes place for values of Weεslightly smaller than 1. The values of θadv, needed to calculate the value of αin (3.5), (3.6)and(3.12), are provided in either the original references or in tables 1 and 2. The experimental values of the critical Weber number for splashing, corresponding to the condition Weε1, have been represented in (b) as a function of and in (c) as a function of /cos θ0. Panel (c) shows that the experimental data follow the prediction in (3.11a,b), Wec∝(/ cos θ0)−3/5. The black symbols in (a–c) indicate additional experiments carried out here using mixtures of water and ethanol for different values of the ethanol mass fraction (in %). The material properties of the mixtures as well as the corresponding values of the contact angles are indicated in table 2. The acronym WE followed by a number indicates the proportion of ethanol in the water–ethanol mixture. The splash velocities predicted by (3.5)and(3.12) using the measured values of θadv given in tables 1 and 2are compared with the experimental values corresponding to either rough or smooth substrates with different wettabilities and different liquids in figures 19 and 20(a), validating our approach. Interestingly, (3.5)and(3.12) predict that the splash 917 A50-22 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press Spreading and splashing of drops impacting rough substrates Surface Liquid θθ 0θadv P220 SC SP Water 45 ±18 ◦71 ±8◦— P220 SC SP WE 05 % 55 ±10 ◦79 ±5◦— P220 SC SP WE 20 % 33 ±3◦70 ±7◦— P220 SC SP WE 60 % 3 ±1◦SHydrophilic ∼0◦— P220 SC SP Ethanol ∼0◦SHydrophilic ∼0◦— P500 SC SP Water 65 ±3◦82 ±3◦— P500 SC SP WE 05 % 50 ±12 ◦77 ±5◦— P500 SC SP WE 20 % 36 ±12 ◦67 ±12 ◦— P500 SC SP WE 60 % 2 ±1◦SHydrophilic ∼0◦— P500 SC SP Ethanol ∼0◦SHydrophilic ∼0◦— P1000 SC SP Water 74 ±1◦85 ±2◦— P1000 SC SP WE 05 % 68 ±5◦83 ±3◦— P1000 SC SP WE 20 % 52 ±7◦77 ±4◦— P1000 SC SP WE 60 % 3 ±1◦SHydrophilic ∼0◦— P1000 SC SP Ethanol ∼0◦SHydrophilic ∼0◦— SC SP Water — 75 ±12 ◦— SC SP WE 05 % — 78 ±7◦— SC SP WE 20 % — 68 ±13 ◦— SC SP WE 60 % — ∼0◦— SC SP Ethanol — ∼0◦— Smooth Glass Water — — 32 ◦ Rough Glass (P4000) Water — — 49 ◦ Rough Glass (P500) Water — — 20 ◦ Parafilm Water — — 109 ◦ Smooth PlexiGlass Water — — 76 ◦ Rough PlexiGlass (P4000) Water — — 89 ◦ Rough PlexiGlass (P500) Water — — 146 ◦ Smooth Glass Ethanol — — ∼0◦ Smooth Glass Methanol — — ∼0◦ Smooth Glass Acetone — — ∼0◦ Smooth Glass Decamethyltetrasiloxane — — ∼0◦ Smooth Glass Dodecamethylpentasiloxane — — ∼0◦ Smooth Glass Isopropanol — — ∼0◦ Smooth Glass 1:4 (v/v) Glycerol/water — — 32 ◦ Stainless Steel Water — — 84 ◦ Stainless Steel Ethanol — — ∼0◦ Table 2. Values of the contact angle θ, of the Young contact angle θ0and of the advancing contact angle θadv for drops of different liquids over different types of substrates. The values of θ0have been calculated using the measured values of θtaking f=3infigure 3. With the purpose of comparing our predictions with the measured splash velocities reported by Range & Feuillebois (1998), Tang et al. (2017)andHao(2017), the values of the advancing contact angles for stainless steel, glass and Plexiglas substrates, which can be either smooth or roughened using the type of sandpaper indicated in parenthesis, are also given in the table. The values of the static advancing contact angle for the liquids used in Riboux & Gordillo (2014) and Palacios et al. (2013) are also provided here. The material properties corresponding to the different water–ethanol (WE) mixtures used here and characterized by the ethanol mass fraction, are the following (de Goede et al. 2021): (i) 5 % of ethanol, ρ=989.0kgm −3,σ=56.4mNm −1,μ=1.23 ×10−3Pa s, (ii) 20 % of ethanol: ρ=968.7 kg m−3,σ=38.0mNm −1,μ=2.14 ×10−3Pa s, (iii) 60% of ethanol, ρ=891.1kgm −3,σ=26.2mN m−1,μ=2.55 ×10−3Pa s. threshold velocity decreases when θadvincreases. A similar result was very recently reported by Quetzeri-Santiago et al. (2019a,b) who, however, quantified the effect of the substrate wettability on the splash transition by modifying the value of the parameter 917 A50-23 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press P. García-Geijo, E.S. Quintero, G. Riboux and J.M. Gordillo βin Riboux & Gordillo (2014) and Gordillo & Riboux (2019). Therefore, the results in figure 20(a) contrast with those in Range & Feuillebois (1998) for the case of drops impacting over smooth Plexiglas substrates and also with those in de Goede et al. (2018) for the case of drops of water–ethanol mixtures impacting over smooth parafilm substrates. Indeed, Range & Feuillebois (1998) and de Goede et al. (2018) find that the splash threshold velocity is mostly independent of the substrate wettability whereas the results in (3.5)and(3.12)andfigures 19 and 20(a) reveal that larger values of θadvfavour the splash transition. The reason for the discrepancies found with previous results lies on the fact that the disintegration process for the case of water drops takes place at very small time and length scales, which are only observable using the adequate spatial and temporal resolutions, as it is shown in figures 20(c)–20(d). Figures 19 and 20(b) also confirm that the solution of (3.1), (3.2)and(3.7)withλ=1 can be used to predict the value of the critical Weber number for the case of non-wetting rough substrates. But, possibly, the most interesting result shown in figure 19 is that the second of the equations in (3.11a,b), which depends on the static contact angle but which does not depend on the Ohnesorge number provided that Oh −1/21, can be used to predict the splash transition when the liquid wets the substrate and Weε1 namely, when the thickness of the lamella is similar or smaller than the grit size. The results obtained here for Weε1 have been confirmed in a separate and independent study by de Goede et al. (2021) for different liquids and values of Oh. Notice that figure 19 also includes the experimental results corresponding to different water–ethanol mixtures, with values of the Young angle θ0given in table 2. The good agreement between the calculated and the measured splash velocities provide further support to the second of the equations in (3.11a,b), which can then be used to predict the value of Wecwhen the liquid wets the substrate and Weε1. Let us point out that the comparison between predictions and experiments in figure 19 reveals that the value of the critical Weber number corresponding to the case of AO sandpaper with =0.7×10−3is well above our predicted value for non-wetting substrates. The reason for this particular behaviour was already pointed out above and it is clearly depicted in figures 7 and 9: the rim is not clearly separated from the solid in this case. But, what is the reason this specific sandpaper behaves so differently with respect to the rest of the cases investigated? To answer this question, let us turn back to the findings in Roisman et al. (2015), where it was suggested that droplet splashing is not controlled by the height of the substrate roughness but by a parameter measuring the slope of the corrugations: in fact, this is not the case because the results in figure 19 for Weε>1 confirm the essential role played by the height and width of the corrugations in triggering the splashing of droplets. However, the observation made by Roisman et al. (2015), together with the results depicted in figure 19, suggest that the splashing of droplets on rough substrates when θ<90 ◦depends on and also on the dimensionless parameter measuring the slope of the corrugations defined in (2.2). Indeed, the values of Sdq in table 1 reveal that the value of the slope of the corrugations for the case of the lime AO sandpaper with =0.7×10−3is far smaller than for the rest of the cases, with a value of the static contact angle θ<90 ◦. Then, for a given value of , two different values for the splash velocity are possible depending on whether the advancing rim wets the substrate, which happens for Sdq 1, or not. This dual behaviour of the rough substrate, which manifests itself through two different values of Wecfor the same value of , can be clearly seen in figure 19 for Weε1. As a final remark, please recall that the results presented here have been deduced for the case in which neither the spatial distribution nor the geometry of 917 A50-24 https://doi.org/10.1017/jfm.2021.313 Published online by Cambridge University Press 48 Capítulo 4. Role of liquid viscosity and of air entrapped on the splashing of drops... de splash propuesto por Quintero, Riboux & Gordillo [42] una vez que los esfuerzos cortantes del flujo en la pared son expresados como una función decreciente del factor de fricción λo, lo que es lo mismo, de la proporción de aire atrapado en el sustrato. Paper reference: García-Geijo, P., Riboux, G., & Gordillo, J. (2022). Role of liquid viscosity and of air entrapped on the splashing of drops impacting over superhydrophobic substrates. Physical Review Fluids,7, 093606. doi:10.1103/PhysRevFluids.7.093606 Physical Review Fluids: Impact Factor: 2.895 (JCR 2021) Category: Physics, Fluids & Plasmas Category Rank: 14/34 (JCR 2021) Category Quartile: Q2 (JCR 2021) Scan the QR code for accessing paper in the publisher website: 5 Conclusions When a drop impacts a solid substrate it will either spread tangentally to the surface or, if the impact velocity is large enough, it will splash disintegrating into tiny droplets which are ejected radially outwards. In this thesis we have extended the ideas in previous works [16, 37,38, 42] and we have also presented new theoretical and experimental results in order to account for the influence of substrate inclination, wettability and roughness on the processes of spreading or in the transition from spreading to splashing under normal atmospheric conditions. Our main contributions in this thesis can be summarized as it is detailed below. 5.1 Spreading In contrast with the case of normal drop impact, the time evolving shapes developed when a drop impacts on inclined surfaces are asymmetric since the drop elongates in a preferential direction as it expands over the surface. The spreading of a drop on an inclined substrate has been described here by including the influence of the angle of inclination χ in the theoretical model previously reported by Ref. [16] where the drop expansion process when the liquid impacts normally to a solid wall is addressed. The angle formed between the impact direction and the normal vector to the substrate possesses a clear effect on the boundary layer flow, which is not axisymmetric as in the normal impact case. Yet, the greatest effect of the inclination angle on the drop spreading proccess relies on the pinning of the advancing contact line over the solid, which does not take place simultaneously for the different angular positions along the drop perimeter. Indeed, this effect is the main reason explaining the asymmetries observed during drop spreading on an inclined substrate. Thus, the asymmetries observed when a drop impacts over solids with different inclination angles is solved by applying balances of mass and momentum at the rim based on the ideas reported in Ref. [16]. The resulting theory provides analytical expressions that accurately predict the experimental results for the temporal evolution of the drop asymmetric shape from the instant when the drop touches the solid until the moment 49 50 Capítulo 5. Conclusions Figura 5.1 Superimposed experimental images of the water drop spreading shapes at different instants of time: t=TV /R=0,2,4 and 6 , impacting at approximately the same value of Weber number, We =ρ(Vcosχ)2R/σ=120±4 , onto inclined glass slides that form an angle (a)χ=15◦ , (b)χ=30◦ and (c)χ=45◦ with the horizontal. The solid coloured lines correspond to the solution of the proposed spreading model (see Chapter 2). The impact point is indicated with a cross. when the rim sticks to the surface, t∼6 (see figure 5.1). At this point, the rim remains static and the drop attains its final shape. Indeed, the liquid stain that remains on the wall, possesses a shape that can be approximated to that of an ellipse whose major and minor semiaxes can be predicted using the algebraic equations deduced in Chapter 2. Through these simple expressions, which are obtained by further simplifying the above mentioned spreading model, we are able to reproduce fairly well the final shape of the drop (see figure 5.2). Moreover, these equations are still valid even at larger dimensionless times, of the order t∼103 , for which gravitational effects can no longer be neglected: in fact, our algebraic equations reproduce fairly well the contour of the stain exceptuating the long draining rivulet which develops at the bottom part of the drop as a consequence of gravity. These results are of great interest for applications related with ink-jet printing, where the maximum expansion radius of the drop needs to be determined as a function of the control parameters. Furthermore, substantial information, such as the drop impact velocity or the impact direction, can be inferred from the analysis of the final shape of the drop thanks to our closed algebraic expressions, which therefore possess potential applications in forensic sciences. 5.2 Splashing When the impact velocity is large enough, the drop disintegrates into smaller pieces (droplets) which are ejected radially outwards from the impacting region. As it was pointed out in the introduction, the critical velocity for splashing depends on the substrate roughness and wettability, with these surface properties being related with each other. Figure 5.3 5.2 Splashing 51 Figura 5.2 Experimental images corresponding to the stain left by a water drop impacting on a surface with two different values of the inclination angle for different values of the Weber number, We =ρ(Vcosχ)2R/σ : (a)χ=15◦We =36 , (b) χ=15◦We =103 , (c)χ=15◦We =163 , (d)χ=30◦We =29 , (e)χ=30◦ We =82 , and (f)χ=30◦We =132 . Blue lines correspond to the solution of the algebraic equations deduced in order to determine the final shape of the drop after spreading (see Chapter 2). All the experimental images correspond to approximately the same instant of time, t=TV /R=10±0.09 , when the rim is already pinned to the surface and the final drop shape has been attained. The impacting point is indicated with a cross. summarizes the main contributions in this thesis since it characterizes, for a given value of the Ohnesorge number, the different types of splashing depending on two parameters, namely, the substrate wettability, measured through the value of the static contact angle θ0 , and also the relative substrate roughness, ε/R . Notice that, for the case of hydrophobic substrates, θ0>π/2 , the splash transition depends on the friction factor λ , which is a function of the amount of air entrapped between the liquid and the substrate and, hence, it is a function of the amplitude of the surface corrugations, as it is detailed in Chaper 4. 5.2.1 Hydrophilic splashing The case of partially wetting substrates, when θ0<π/2 which corresponds to the pink region in figure 5.3, can in turn be divided in two different regimes depending on the 52 Capítulo 5. Conclusions Figura 5.3 Drop splashing at normal atmospheric conditions can be classified in three main regimes: for the case of hydrophilic substrates, the critical conditions for splashing depends on the value of the parameter Weε=ρV2ε/σ . If Weε<1 and θ0<π/2 , the edge of the expanding lamella needs to lift off from the surface first for splashing to occur and, hence, the spreading-splashing transition can be predicted using the theory in Refs. [37,38] which corresponds to the cases of smooth partially wetting substrates once the new equation for α=α(θadv) proposed here (see Chapter 4) is employed. But, if the solid is hydrophilic and Weε>1 , drop splashing occurs when the liquid velocity in the lamella is larger than the velocity at which the liquid wets the corrugations. For the case of hydrophobic substrates, θ0>π/2 , the rim does not contact the substrate and the splash criterion changes to the one deduced in Ref. [42], which depends on the time evolution of the thickness of the rim that borders the expanding lamella. This thickness depends on the friction between the liquid and the solid through the dimensionless friction factor λ [16], whose value decreases as the amount of air entrapped between the drop and substrate increases, as it is detailed in Chapter 4. 5.2 Splashing 53 value of the parameter Weε=ρV2ε/σ firstly identified here and which expresses the ratio between the height of the corrugations, ε , and the thickness of the ejected lamella (see Chapter 3). When Weε.1 , drop splashing is limited by the condition put forward in [37] for the case of smooth hydrophilic surfaces once the influence of the static advancing contact angle, θadv , is taken into consideration in the value of the wedge angle α , see Figure 5.3. This splashing criterion is based on the idea that the rim needs to be separated first from the substrate with a vertical velocity that needs to be larger than the velocity of radial growth of the rim caused by capillary retraction. The vertical velocity needed for the rim to lift off from the wall is induced by the air lubrication force at the wedge region of angle α developed between the advancing lamella and the solid: indeed this aerodynamic force is proportional to the product of the relative velocity between the rim and the surrounding atmosphere and 1/tan2α , being α a function of the static advancing contact angle, which increases with the substrate rugosity. Indeed, as it was pointed out in Ref. [38], α∼60◦ and its value slightly varies with wetting properties of the surface, with larger values of α and, hence, smaller values of the lift force, for smaller values of θadv . Let us emphasize here that the proposed empirical equation for α as a decreasing function of the static advancing contact angle presented here for the first time (see Chapter 3) together with the term 1/tan2α , explains why the critical velocity for splashing in corrugated substrates is smaller than for the case of smooth substrates when θ0<π/2and Weε.1. However, when the amplitude of the hydrophilic substrate corrugations is larger than the thickness of the lamella namely, Weε&1 , the previous splash criterion is no longer valid because the width of the lamella is similar or even smaller than the height of the surface asperities. Indeed, under these conditions, a lamella is not developed, but the liquid flows along channels of characteristic width ε (see figure 5.3). Thus, splashing takes place when a jet of typical diameter ε is formed and the speed at which this jet is ejected, is larger than the velocity at which the liquid wets the channels formed by the corrugations of width ε. 5.2.2 Hydrophobic splashing Finally, for the case of drops impacting superrepelent surfaces, θ0>π/2 , which corresponds to the blue region in figure 5.3, it is observed that the advancing rim is never in contact with the substrate, regardless of the substrate roughness. Based on this observation, Ref. [42] proposed a model to predict the splash transition for the case of superhydrophobic surfaces which notably differs from the criteria presented for hydrophilic substrates because, in this case, drop splashing is limited by the growth rate of the rim thickness. Indeed, as it was discussed at length in Ref. [45], the edge bordering the expanding lamella disintegrates into tiny droplets only if capillary disturbances grow faster than the rim thickness b(t) (see figure 5.3). Experiments also reveal that the value of the critical Weber number for splashing, We∗ , decreases as the wall shear stress is reduced. Indeed, an increase of either the friction factor, λ , or of the Ohnesorge number, yields to larger values of We∗ . Moreover, the amount of air pockets present between the impacting drop and the wall clearly increases with the amplitude of the substrate asperities. Therefore, when surface roughness is higher, the value of the friction factor decreases because of the larger proportion of air entrapped in the corrugations. Motivated by these facts, λ 54 Capítulo 5. Conclusions is expressed in this thesis just as a decreasing function of ε through an empirical linear piecewise relationship (see Chapter 4), thus extending the previous results reported in Refs. [42,45]. 5.3 Concluding remarks In conclusion, the spreading-splashing transition can be accurately predicted making use of the equations deduced in this thesis, which extend the previous results in Refs. [16,37,38,42, 45] by explicitly including in the resulting expressions the influence of both the static contact angle and of the substrate roughness. The theoretical models put forward in this thesis compare favourably with the experimental measurements corresponding to liquids of different viscosities impacting over substrates with varying wettabilities and rugosities. 5.4 Outlook Future work should focus on analysing the spreading of liquids with higher viscosities over inclined substrates and also the effect of viscosity and roughness on the splash transition when the drop impacts obliquely over the wall. 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