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Project titleComprehensive study of heat and mass transfer in evaporative lithography, physicochemical and geometric parameters of the resulting functional coatings: experiment, theory and modeling

Research area 09 - ENGINEERING SCIENCES, 09-711 - Methods of nano-structuring (nano-lithography and accompanying processes)

Keywordsfunctional coatings, micro- and nanostructures, evaporative lithography, droplets and films, capillary flows, Marangoni effect, colloidal particles, phase transitions

Annotation
Intensive development of technologies is impossible without the development of new manufacturing methods. Currently, the creation of miniature devices, the production of materials and coatings with different functional properties, and the development of software for modeling technological processes in order to optimize them are relevant areas in the technological and economic development of our society. All over the world, there is serious competition in these areas, reminiscent of the evolutionary processes in nature. If you do not focus on some new and promising direction in time and do not show perseverance in your work, you can hopelessly fall behind. This often requires large human, technical, financial, and material resources. The development of micro-and nanostructured coatings is important for various fields of activity: micro-and optoelectronics, biotechnology, medical diagnostics, nanotechnology, and others. Topographically structured colloidal sediments and solid films, for example, are associated with applications such as functional coatings, photonic crystals, transparent flexible conductive coatings, inkjet printing, laboratories on a chip, biosensors, and miniature medical devices. The number of such applications is large and continues to grow. There are different methods of obtaining such structures: self-organizing methods (for example, evaporation-induced self-assembly), lithographic approaches (photolithography, nanoimprint lithography, capillary lithography, nanosphere lithography, etc.), and other technologies (for example, spin coating, dip coating, die coating, Langmuir–Blodgett method, etc.). Each of these methods is suitable for different problems and is characterized by certain advantages and disadvantages. A relatively new and promising method of forming structured sediments and coatings, which is dedicated to the claimed project, is called evaporative lithography. It is not at all expected that in the future evaporative lithography will replace the methods mentioned above. Most likely, it will harmoniously complement them, since it is preferable in some cases. Evaporative lithography is a method based, as a rule, on a single-stage process. It does not require complex and expensive equipment and is applicable to various materials and surfaces without pretreatment. Despite the fact that the first ideas and modest attempts in this direction appeared about 20 years ago [Routh & Russel, AIChE J, 1998; Deegan et al., Phys. Rev. E, 2000], and more detailed experiments and mathematical models on some variations of the method about 10 years ago or less ([Harris et al., Philos Trans R Soc A Math Phys Eng Sci, 2009; Parneix et al. Phys Rev Lett, 2010; Georgiadis et al., Soft Matter, 2011; Salas et al., J Phys ChemC, 2012; Arshad & Bonnecaze, Nanoscale, 2013; Vodolazskaya & Tarasevich, Eur Phys J E, 2017; Kolegov, Microgravity Sci Technol 2018], etc.), it is necessary to do even more work before this method is widely and actively used in technological production. In this relatively short period of time (1998-2021), there is a transformation of basic research into applied research, from the explanation of the coffee ring effect to the development of specific applications. Our recent analysis [Kolegov & Barash, Adv. Colloid Interface Sci., 2020] allowed us to identify the main achievements and problems in the field of evaporative lithography, suggest further possible solutions, and focus on specific work. Now one of the problems is as follows. The evaporative lithography is inferior to some methods of self-organization caused by evaporation (for example, based on periodic movement and pinning of the edge boundary), according to such characteristics as the spatial resolution of the formed pattern. On the other hand, it provides more possibilities in terms of the variety of geometric shapes of the resulting structures. Thus, one of the directions of further work should be associated with reducing the size of individual periodically repeated elements of structures. In our opinion, it is necessary to move along the path of hybridization of methods in order to solve this problem. Another problem is related to the small area of formed sediments and coatings. For some applications, topographically structured surfaces with an area measured in square millimeters or centimeters are not sufficient. Additional numerical and experimental studies are also needed in this direction. Another field of activity is the study of new ways to control nonuniform evaporation in space and time, effective and simple from the point of view of industrial implementation. The scientific novelty of our research will be associated with the construction of more complex mathematical models that will take into account an even greater number of different effects and more accurately describe the processes of heat and mass transfer during inhomogeneous evaporation of droplets and films containing dissolved and suspended matter, and with performing experiments and developing new modifications of this method. We are talking about such effects as capillary flows, Marangoni flow, heat transfer in three phases (air, liquid, solid), mass transfer in two phases (liquid and air), Rayleigh–Benard flows, the effect of the local concentration and temperature of the solution on the hydrodynamics, changes in the viscosity of the solution. It is also planned to consider multicomponent solutions (with sets of particles of different shapes, sizes, and materials), particle interaction forces (capillary and electrostatic), different modes of the three-phase ("liquid–substrate-air") boundary (fixing the contact line, the movement of this boundary, and the periodic change of these two modes), different materials of substrates and cells, types of liquids with different physical and chemical parameters, phase transitions (liquid-vapor, sol-gel, solution-glass, etc.), take into account mobile boundaries ("liquid–air", "sediment–liquid") and much more. Our research will cover various modifications of evaporative lithography, which will allow us to make a detailed comparison and analysis of their capabilities. Different approaches to the control of particle deposition, liquid flows, particle sintering, and other processes will be considered (the use of composite substrates with periodically varying parameters, such as thermal conductivity; local heating of the liquid, such as light or due to rod heaters mounted in the substrate; changing the vapor concentration in local areas using airflow sources, and much more). These problems are computationally expensive due to their complexity. It is planned to use not only existing software packages for modeling but also to develop software for some individual problems, taking into account their specifics. The modification of some numerical methods, the development of new efficient algorithms for high-performance computing on supercomputers (including hybrid ones), and the creation of software for modeling are among the priority areas. Simulations will allow us to identify the key parameters and their ranges of values for effective process management and obtaining the structures of the necessary shape, square, and resolution.

Expected results
The simulation results will allow us to explain the mechanisms of the formation of micro-and nanostructures in the process of inhomogeneous drying of colloidal droplets and films. Parameters of the system allowed one to control the process and to obtain the sediments will be determined based on the results of numerical calculations. Experimental studies will be carried out and various methods of controlling structure formation in evaporative lithography will be considered. This direction is currently the mainstream. Many scientific groups from different countries are working intensively in this direction. Our analysis shows that theoretical and experimental research is developed in this direction relatively slowly. Thus, the results of this work will correspond to the world level. During the project, practical recommendations will be developed. It will be able to be used to create functional coatings using composite substrates with not constant temperature properties and nonuniformly heated surfaces, external air, and light sources, as well as using various hybrid approaches. This is important for creating functional coatings. The introduction of these methods into production will allow us to create new materials in a more efficient way. Modifications of the numerical methods will be developed taking into account the specifics of the described problems. New parallel algorithms will also be implemented to make efficient use of supercomputer equipment and obtain simulation results in a reasonable time. Open libraries and modules for numerical modeling with a detailed technical description can be further adapted by other researchers for similar problems. Besides, in the future, the developed software will be useful for the scientific community and engineers.

Annotation of the results obtained in 2022
A modification of the numerical method is proposed, in which splitting by physical processes, an iterative method of explicit relaxation and Thomas algorithm are combined. A practical recipe for suppressing sawtooth oscillations is described using the example of a specific problem. A software module has been developed for numerical simulation of hydrodynamics in droplets and films in C++, which can be used in the future for problems of evaporative lithography. With the help of this module, numerical calculations were carried out, the results of which were compared with the results obtained in the Maple package. Numerical simulation predicted the case when the direction of the capillary flow in a drop drying on a substrate changes to the opposite over time due to a change in the sign of the gradient of the vapor flux density. This can lead to a slowdown in the removal of the substance to the periphery, which as a result will contribute to the formation of a more or less uniform precipitation over the entire contact area of the droplet with the substrate. This observation is useful for improving methods of suppressing ring precipitation associated with the effect of coffee rings and undesirable for some applications, such as inkjet printing or coating. The preprint of the article can be found at https://arxiv.org/abs/2301.06983 as well as with the software module, the code of which is publicly available in the Fund of Algorithms and Programs of the SB RAS (https://fap.sbras.ru/node/5114 ). Flows in a flattened drop suspended on a round frame are theoretically investigated. Marangoni convection occurs due to a vertical temperature gradient associated with external action across the droplet and is caused by changes in surface tension at the free boundary. Using an analytical approach for solutions of the Stokes equation in the coordinates of an oblate spheroid, linearly independent stationary solutions for Marangoni convection are obtained in terms of Stokes stream functions. On the basis of numerical simulation of thermocapillary motion in droplets, the occurrence of a stationary regime is investigated. Both analytical results and numerical calculations predict axisymmetric circulating convective motion in the droplet, the dynamics of which is determined by the magnitude of the temperature gradient in the transverse direction. Analytical solutions for the critical temperature distribution and velocity field are obtained for large temperature gradients across the flattened droplet. These solutions show the lateral separation of critical and stationary movements inside the droplets. Critical vortices are localized near the central part of the droplet, and an intense stationary flow is closer to its end. The transition to the flat film limit is investigated within the framework of the application of current functions by reducing the ellipticity coefficient of droplets to zero. The initial stationary regime for strongly flattened droplets becomes unstable with respect to multi-vortex perturbations by analogy with flat liquid films with free boundaries. The preprint of the article is available at https://arxiv.org/abs/2304.14512 Published article: Vodolazskaya I.V. Simulation of the influence of the physical properties of the liquid and the substrate on the direction of flows in a drop evaporating from a microcell // PERM HYDRODYNAMIC SCIENTIFIC READINGS: A collection of articles based on the materials of the VIII All-Russian Conference, October 5-7, 2022. Perm State National Research University, 2022. - P. 124-130, https://www.elibrary.ru/item.asp?id=49953735 . In this paper, we propose a physical model that allows us to study the influence of thermal effects arising from cooling during liquid evaporation on hydrodynamic flows in drops placed in micrometer-sized cylindrical cells. The influence of the physical properties of the substrate and liquid on the direction of flows is investigated. The mathematical model is based on the laws of conservation of matter and energy, the heat conduction equation and the equation of fluid dynamics in the lubrication approximation. The numerical calculation of the model, performed using the FlexPDE package, made it possible to determine the boundaries of the dimensionless parameters of the model, on which the flow direction changes. Mathematical modeling of hydrodynamics in a liquid drop evaporating from an open cylindrical cell of micrometer size was carried out. The developed model makes it possible to investigate the influence of thermal effects arising from cooling during liquid evaporation on hydrodynamic flows in a drop. We take into account the dependence of the surface tension of the liquid on temperature. The model describes vapor diffusion in the air, heat distribution in the cell and in the liquid due to thermal conductivity, thermocapillary fluid flow, movement of the two–phase liquid-air boundary and compensatory fluid flow resulting from evaporation. The mathematical model is based on the laws of conservation of matter and energy, the vapor diffusion equation, as well as the equation of fluid dynamics within lubrication approximation in combination with the kinematic approach. The calculation results of the flow velocity in a drop of ethylene glycol according to the obtained analytical formula are in good agreement with the experimental literature data. The article “Vodolazskaya I.V. Simulation of hydrodynamics in a liquid evaporating from a cylindrical microcell using the thin layer approximation and the kinematic approach” has been accepted for publication in the journal "Computational Continuum Mechanics". A mathematical model describing the evaporation of a liquid film consisting of a polymer dissolved in alcohol (methanol–poly(vinyl acetate) solution with 67 wt.% methanol) on a composite substrate has been developed. An aluminum substrate is considered, on which a layer of Teflon is applied in the form of a strip. Thus, the thermal properties of the substrate are spatially inhomogeneous. The model is based on the lubricant approximation and takes into account capillary flows, solute diffusion, thermal and solutal Marangoni flows, liquid evaporation (intensity depends on temperature), sol–gel phase transition and spatial heterogeneity of the thermal conductivity coefficient of the substrate. The results of the test calculations made it possible to conclude that in this system, the dominant factor affecting mass transfer and the formation of a solid relief coating during the evaporation of methanol is the solutal Marangoni flow, which confirms the hypothesis of the authors of the experiment [Cavadini P. et al. Chem. Eng. Process. 2013, http://dx.doi.org/10.1016/j.cep.2012.11.008]. The numerically predicted shape of the relief polymer coating is in qualitatively agreement with their experimental measurements. A study of the deposition of polystyrene particles in cells with different geometries has been performed. The curvature of the free surface of the liquid depends on the geometry of the cell, as well as the local inhomogeneity of the vapor flow density. In the process of evaporation, capillary flows arise, which affect the transfer and deposition of colloidal particles. Cells with sides in the form of a circle, square and triangle were made for conducting experiments. Cases with water and isopropyl alcohol are considered. The contact angles for the rim and the substrate were measured, and the profiles of the meniscus (the shape of the free surface of the liquid in an open cell) were obtained. It is shown that the shape of the sediment is influenced not only by the geometry of the cell, but also by the volatility of the liquid. The effect of uneven heating of the cell and its shape on the geometric characteristics of the collected cluster of microparticles has been experimentally investigated. The temperature gradient is created due to the heating element mounted in the substrate. This leads to a gradient of surface tension, and, as a consequence, to the appearance of a thermocapillary flow, which carries the particles.

Publications

1. Vodolazskaya I. V. Моделирование влияния физических свойств жидкости и подложки на направление течений в капле, испаряющейся из микроячейки ПЕРМСКИЕ ГИДРОДИНАМИЧЕСКИЕ НАУЧНЫЕ ЧТЕНИЯ Сборник статей по материалам VIII Всероссийской конференции, посвященной памяти профессоров Г. З. Гершуни, Е. М. Жуховицкого и Д. В. Любимова (г. Пермь, ПГНИУ, 5–7 октября 2022 г.)., Пермский государственный национальный исследовательский университет Институт механики сплошных сред УрО РАН: Издательский центр Пермского государственного национального исследовательского университета 614990, Пермь, ул. Букирева, 15, 2022.— С. 124-130 (year - 2022)

2. Vodolazskaya I.V. Моделирование гидродинамики в испаряющейся из цилиндрической микроячейки жидкости с использованием приближения тонкого слоя и кинематического подхода Вычислительная механика сплошных сред, - (year - 2023)

3. - Программа для моделирования массопереноса в высыхающей на подложке капле в приближении тонкого слоя -, PR23002 (year - )

4. - Программа для моделирования формирования осадка в виде монослоя бинарной смеси микрочастиц разного размера при высыхании капли на гидрофильной подложке -, 2022665336 (year - )

Annotation of the results obtained in 2023
The case is investigated when Marangoni convection occurs due to a vertical temperature gradient across a droplet and is caused by changes in the surface tension on the free surface of the droplet. When considering the Stokes equation in the coordinates of an oblate spheroid, we analytically obtained linearly independent stationary solutions for Marangoni convection in terms of the stream function. Numerical simulation of thermocapillary motion in droplets was used to study the onset of a stationary regime. Both analytical and numerical calculations predict axisymmetric circulating convective motion in a droplet, the dynamics of which is determined by the magnitude of the temperature gradient across the droplet. Analytical solutions for the critical distribution of temperature and velocity fields are obtained for large temperature gradients across the flattened droplet. These solutions show the lateral separation of critical and stationary motions inside the droplets. Critical vortices are localized near the central part of the droplet, while an intense stationary flow is located closer to its end. The article has been published: Shishkin M. A., Kolegov K. S., Pikin S. A., Ostrovskii B. I., Pikina E. S. Marangoni instability in oblate droplets suspended on a circular frame // Physics of Fluids.— 2023.— 07.— Vol. 35, No. 7.— P. 077109. A mathematical model of mass transfer in a droplet drying on a substrate based on the lubrication approximation is described. The model takes into account the transfer of dissolved or suspended matter by capillary flow, diffusion of this substance, evaporation of liquid, formation of solid sediment, dependence of viscosity and density of vapor flux on the concentration of impurity. The case is considered when the three-phase boundary “liquid-substrate-air” is pinned. Explicit and implicit difference schemes have been developed for the equations of the model. A modification of the numerical method is proposed. An open source software module has been developed. The article has been published: Kolegov K. S. “Suppression of sawtooth oscillations when using a finite-difference scheme for mass transport simulation in a drying droplet on a substrate in the thin layer approximation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 27:2 (2023), 309–335. A model has been developed that makes it possible to study the effect of thermal effects in a liquid due to its cooling during evaporation on hydrodynamic flows when placed in a micrometer-sized cylindrical cell. The article has been published: Vodolazskaya I. V. Simulation of hydrodynamics in a liquid evaporating from a cylindrical microcell using the thin layer approximation and a kinematic approach // Computational Continuum Mechanics, 2023, 16 (3), 368-374. The solution of the problem of evaporation of a film consisting of a polymer dissolved in alcohol on a composite substrate has been completed. A mathematical model is proposed that allows us to explain the basic mechanisms of formation of thin polymer films (height < 50 microns) during the evaporation of alcohol from a solution. The thermal conductivity of the substrate is spatially uneven, which leads to inhomogeneous evaporation along the free surface of the film. After drying, a relief polymer coating remains on the substrate. It is shown that solutal Marangoni flow plays a primary role in the process under consideration. The article has been published: Kolegov K. S. Simulation aspects of patterning polymer films via evaporative lithography and composite substrates // Physics of Fluids.— 2024.— April.— Vol. 36, Iss. 4.— P. 042002. A mathematical model describing the transfer of nanotubes in a droplet evaporating on a substrate has been developed. The effect of the particle charge, the time of complete evaporation of the droplet (capillary flow velocity) and particle diffusion on the sediment morphology has been numerically investigated. An information system has been developed to analyze the morphology of colloidal assemblies. Web technologies and a database management system were used in the development. Access to the Information System for Analysis of Nanostructure Morphology platform is available online at https://isanm.space / . It allows you to store and process numerical and experimental data, save processing results, and import / export data. This system is a useful tool for specialists working in the field of evaporative lithography, photonic crystals, evaporative self-assembly, colloidal chemistry, nanosphere lithography, and other related fields. The drying of a drop of pure water on a salt substrate has been studied when several processes occur simultaneously: dissolution of a solid, evaporation of a liquid, transfer of salt ions by a flow, as well as the nucleation and growth of a salt deposit near the periphery of the drop. The final salt deposit on the substrate is a thin shell forming either inclined walls or a hollow annular rim. The proposed model made it possible to explain the main patterns characterizing the drying of a drop of pure water on a soluble substrate – a single crystal of salt. Initially, the appearance of a concentration gradient inside the liquid due to the dissolution of the salt leads to the appearance of solutal Marangoni flow. The progressive saturation of the drop with salt leads to a stop of the Marangoni flow, over which the capillary flow begins to prevail. The dried drop leaves an annular hollow spot on the substrate, which resembles a longitudinal section of a torus. The study shows that the morphology obtained is quite stable and characteristic of a wide range of parameter values: the radius of the base of the sessile droplet, the contact angle, temperature, and relative humidity. The article has been published: Mailleur Alexandra, Pirat Christophe, Rivire Charlotte, Vodolazskaya Irina, Colombani Jean, Experimental and numerical study of the evaporation of a pure water drop on a salt surface // The European Physical Journal Special Topics. 2024. DOI 10.1140/epjs/s11734-024-01119-0. Here we present a combined method that enables the production of centimeter-scale microparticle deposits with the desired geometric shape. The method is based on controlling the massive transport of microparticles by thermocapillary flow in a layer of volatile liquid in a cell with borders of the desired geometry. The proposed method is reliable, easy to implement and potentially capable of producing a wide variety of deposits geometries, making it an attractive technique for patterning and modifying surface properties with particles of any type. It is shown that the particle size and wettability of the substrate affect the type of pattern (the case without a heating or cooling source). If the particle deposition time is less than the time of rupture of the liquid layer due to evaporation, then the particles form patterns in the center of the cell due to their transfer by a near-bottom flow. If the time of particle deposition significantly exceeds the time of layer rupture, then the formation of sediment occurs in the "sticl–slip" mode of the contact line formed during rupture in the central part and moving towards the sides. In this case, concentric structures of a given shape are formed.

Publications

1. Kolegov K.S. Подавление пилообразных осцилляций при использовании разностной схемы для моделирования массопереноса в высыхающей на подложке капле в приближении тонкого слоя Вестник Самарского государственного технического университета. Серия «Физико-математические науки», Т. 27, № 2.— С. 309-335 (year - 2023) https://doi.org/10.14498/vsgtu1994

2. Kolegov K.S. Устранение пилообразных осцилляций при использовании разностной схемы для моделирования массопереноса в высыхающей на подложке капле в приближении тонкого слоя Математическое моделирование и биомеханика в современном университете. Тезисы докладов XVII Всероссийской школы. — Ростов-на-Дону: Южный федеральный университет, С. 5-9 (year - 2023)

3. Kolegov K.S. Simulation aspects of patterning polymer films via evaporative lithography and composite substrates Physics of Fluids, Vol. 36, Iss. 4.— P. 042002 (year - 2024) https://doi.org/10.1063/5.0200880

4. Mailleur Alexandra, Pirat Christophe, Rivire Charlotte, Vodolazskaya Irina, Colombani Jean Experimental and numerical study of the evaporation of a pure water drop on a salt surface The European Physical Journal Special Topics, - (year - 2024) https://doi.org/10.1140/epjs/s11734-024-01119-0

5. Shishkin M. A., Kolegov K. S., Pikin S. A., Ostrovskii B. I., Pikina E. S. Marangoni instability in oblate droplets suspended on a circular frame Physics of Fluids, Vol. 35, No. 7. (year - 2023) https://doi.org/10.1063/5.0154444

6. Shishkin M.A., Kolegov K.S., Pikina E.S. Термокапиллярная конвекция в подвешенных на кольцевом креплении сплюснутых каплях Сборник статей по материалам IX Всероссийской конференции ПЕРМСКИЕ ГИДРОДИНАМИЧЕСКИЕ НАУЧНЫЕ ЧТЕНИЯ, посвященной памяти профессоров Г.З. Гершуни, Е.М. Жуховицкого и Д.В. Любимова, Пермь / отв. ред. Т. П. Любимова, 2023.— С. 467-471 (year - 2023)

7. Kolegov K.S. Формирование рельефной пленки при высыхании раствора ПВА в метаноле на композитной подложке VI Международная конференция по коллоидной химии и физико-химической механике, посвященная 125-летию со дня рождения П.А. Ребиндера (Казань, 2023): тезисы докладов. — Казань: ИОФХ им. А.Е. Арбузова – обособленное структурное подразделение ФИЦ КазНЦ РАН, 2023.— С. 153 (year - 2023)

8. Kolegov K.S., Vodolazskaya I.V., Eserkepov A.V. Моделирование испарительной самосборки нанотрубок при высыхании капли на подложке XXI МЕЖДУНАРОДНАЯ КОНФЕРЕНЦИЯ "МАТЕМАТИКА. КОМПЬЮТЕР. ОБРАЗОВАНИЕ", XV ОБЩЕРОССИЙСКИЙ СИМПОЗИУМ С МЕЖДУНАРОДНЫМ УЧАСТИЕМ "БИОФИЗИКА СЛОЖНЫХ СИСТЕМ: ВЫЧИСЛИТЕЛЬНАЯ И СИСТЕМНАЯ БИОЛОГИЯ, МОЛЕКУЛЯРНОЕ МОДЕЛИРОВАНИЕ". — г. Дубна, 22 – 27 января, 2024.— С. 37 (year - 2024)