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

Cover Page

Cite item

Full Text

Abstract

Evaporating droplets and films are used in applications from different fields. Various methods of evaporative self-assembly are of particular interest. The paper describes a mathematical model of mass transfer in a droplet drying on a substrate based on the lubrication approximation. The model takes into account the transfer of a dissolved or suspended substance by a capillary flow, the diffusion of this substance, the evaporation of liquid, the formation of solid deposit, the dependence of the viscosity and the vapor flux density on the admixture concentration.
The case with pinning of the three-phase boundary (“liquid–substrate–air”) is considered here. Explicit and implicit finite-difference schemes have been developed for the model equations. A modification of the numerical method is proposed, in which splitting by physical processes, the 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 in C++ has been developed, which can be used for evaporative lithography problems in the future. 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 in which the direction of the capillary flow 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 transfer 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 annular deposit suppression associated with the coffee-ring effect and undesirable for some applications, such as inkjet printing or coating.

About the authors

Konstantin S. Kolegov

Astrakhan Tatishchev State University; Tyumen State University

Author for correspondence.
Email: konstantin.kolegov@asu.edu.ru
ORCID iD: 0000-0002-9742-1308
SPIN-code: 1872-4975
Scopus Author ID: 57191072458
ResearcherId: T-6123-2017
https://science.asu.edu.ru/index.php/user/9/

Cand. Phys. & Math. Sci., Senior Researcher, Lab. of Mathematical Modeling and Information Technologies in Science and Education; Center for Nature-inspired Engineering

Russian Federation, 414056, Astrakhan, Tatischev st., 20 a; 625003, Tyumen, Lenin st., 25

References

  1. Zang D., Tarafdar S., Tarasevich Yu. Yu., et al. Evaporation of a droplet: From physics to applications, Phys. Rep., 2019, vol. 804, pp. 1–56. EDN: HMEGXP. DOI: https://doi.org/10.1016/j.physrep.2019.01.008.
  2. Kolegov K. S., Barash L. Yu. Applying droplets and films in evaporative lithography, Adv. Colloid Interf. Sci., 2020, vol. 285, 102271. EDN: BEBGWV DOI: https://doi.org/10.1016/j.cis.2020.102271.
  3. Deegan R. D., Bakajin O., Dupont T F., et al. Capillary flow as the cause of ring stains from dried liquid drops, Nature, 1997, vol. 389, no. 6653, pp. 827–829. DOI: https://doi.org/10.1038/39827.
  4. Kim H., Yang J., Kim S., et al. Numerical simulation of the coffee-ring effect inside containers with time-dependent evaporation rate, Theor. Comput. Fluid Dyn., 2022, vol. 36, no. 3, pp. 423–433. DOI: https://doi.org/10.1007/s00162-021-00602-x.
  5. Baba H., Yoshioka R., Takatori S., et al. Transitions among cracking, peeling and homogenization on drying of an aqueous solution containing glucose and starch, Chem. Let., 2021, vol. 50, no. 5, pp. 1011–1014. DOI: https://doi.org/10.1246/cl.210009.
  6. Wang W., Wang Q., Zhang K., et al. On-demand contact line pinning during droplet evaporation, Sens. Act. B: Chem., 2020, vol. 312, 127983. DOI: https://doi.org/10.1016/j.snb.2020.127983.
  7. Saroj S. K., Panigrahi P. K. Magnetophoretic control of diamagnetic particles inside an evaporating droplet, Langmuir, 2021, vol. 37, no. 51, pp. 14950–14967. DOI: https://doi.org/10.1021/acs.langmuir.1c02968.
  8. Al-Muzaiqer M. A., Kolegov K. S., Ivanova N. A., Fliagin V. M. Nonuniform heating of a substrate in evaporative lithography, Phys. Fluids, 2021, vol. 33, no. 9, 092101. DOI: https://doi.org/10.1063/5.0061713.
  9. Du F., Zhang L., Shen W. Controllable dried patterns of colloidal drops, J. Col. Int. Sci., 2022, vol. 606, pp. 758–767. DOI: https://doi.org/10.1016/j.jcis.2021.08.089.
  10. Cedeno R., Grossier R., Lagaize M., et al. Nucleation in sessile saline microdroplets: Induction time measurement via deliquescence-recrystallization cycling, Faraday Discuss., 2022, vol. 235, pp. 183–197. DOI: https://doi.org/10.1039/d1fd00090j.
  11. Perkins-Howard B., Walker A. R., Do Q., et al. Surface wettability drives the crystalline surface assembly of monodisperse spheres in evaporative colloidal lithography, J. Phys. Chem. C, 2022, vol. 126, no. 1, pp. 505–516. DOI: https://doi.org/10.1021/acs.jpcc.1c07098.
  12. Jose M., Mayarani M., Basavaraj M. G., Satapathy D. K. Evaporative self-assembly of the binary mixture of soft colloids, Phys. Chem. Chem. Phys., 2021, vol. 23, no. 12, pp. 7115–7124. DOI: https://doi.org/10.1039/d1cp00440a.
  13. Zolotarev P. A., Kolegov K. S. Monte Carlo simulation of particle size separation in evaporating bi-dispersed colloidal droplets on hydrophilic substrates, Phys. Fluids, 2022, vol. 34, no. 1, 017107. DOI: https://doi.org/10.1063/5.0072083.
  14. Kirner F., Sturm E. V. Advances of nonclassical crystallization toward self-purification of precious metal nanoparticle mixtures, Cryst. Growth Des., 2021, vol. 21, no. 9, pp. 5192–5197. DOI: https://doi.org/10.1021/acs.cgd.1c00544.
  15. Hossain M. T., Gates I. D., Natale G. Dynamics of Brownian Janus rods at a liquidliquid interface, Phys. Fluids, 2022, vol. 34, no. 1, 012117. DOI: https://doi.org/10.1063/5.0076148.
  16. Mustakim M., Anil Kumar A. V. Depletion induced demixing and crystallization in binary colloids subjected to an external potential barrier, J. Phys. Chem. B, 2021, vol. 126, no. 1, pp. 327–335. DOI: https://doi.org/10.1021/acs.jpcb.1c08591.
  17. Nozawa J., Uda S., Toyotama A., et al. Heteroepitaxial fabrication of binary colloidal crystals by a balance of interparticle interaction and lattice spacing, J. Col. Int. Sci., 2022, vol. 608, pp. 873–881. DOI: https://doi.org/10.1016/j.jcis.2021.10.041.
  18. Galy P. E., Guitton-Spassky T., Sella C., et al. Redox control of particle deposition from drying drops, ACS Appl. Mater. Interfaces, 2022, vol. 14, no. 2, pp. 3374–3384. DOI: https://doi.org/10.1021/acsami.1c18933.
  19. Inoue K., Inasawa S. Drying-induced back flow of colloidal suspensions confined in thin unidirectional drying cells, RSC Adv., 2020, vol. 10, no. 27, pp. 15763–15768. DOI: https://doi.org/10.1039/d0ra02837a.
  20. Homede E., Manor O. Deposition of nanoparticles from a volatile carrier liquid, J. Col. Int. Sci., 2020, vol. 562, pp. 102–111. DOI: https://doi.org/10.1016/j.jcis.2019.11.062.
  21. Li D., Chen R., Zhu X., et al. Light-fueled beating coffee-ring deposition for droplet evaporative crystallization, Anal. Chem., 2021, vol. 93, no. 25, pp. 8817–8825. DOI: https://doi.org/10.1021/acs.analchem.1c00605.
  22. Shao X., Hou Y., Zhong X. Intense jet flow with symmetric vortices induced by saline concentration gradient at free surface of a drying saline droplet, Int. Comm. Heat Mass Transf., 2021, vol. 128, 105573. DOI: https://doi.org/10.1016/j.icheatmasstransfer.2021.105573.
  23. Hegde O., Chatterjee R., Rasheed A., et al. Multiscale vapor-mediated dendritic pattern formation and bacterial aggregation in complex respiratory biofluid droplets, J. Col. Int. Sci., 2022, vol. 606, pp. 2011–2023. DOI: https://doi.org/10.1016/j.jcis.2021.09.158.
  24. Corletto A., Shapter J. G. Thickness/morphology of functional material patterned by topographical discontinuous dewetting, Nano Select, 2021, no. 9, pp. 1723–1740. DOI: https://doi.org/10.1002/nano.202000301.
  25. Bayat F., Tajalli H. Nanosphere lithography: the effect of chemical etching and annealing sequence on the shape and spectrum of nano-metal arrays, Heliyon, 2020, vol. 6, no. 2, e03382. DOI: https://doi.org/10.1016/j.heliyon.2020.e03382.
  26. Bhardwaj R., Fang X., Somasundaran P., Attinger D. Self-assembly of colloidal particles from evaporating droplets: role of DLVO interactions and proposition of a phase diagram, Langmuir, 2010, vol. 26, no. 11, pp. 7833–7842. DOI: https://doi.org/10.1021/la9047227.
  27. Tarasevich Yu. Yu., Vodolazskaya I. V., Isakova O. P. Desiccating colloidal sessile drop: Dynamics of shape and concentration, Coll. Pol. Sci., 2011, vol. 289, no. 9, pp. 1015–1023. EDN: OHRMJP. DOI: https://doi.org/10.1007/s00396-011-2418-8.
  28. Kolegov K. S., Lobanov A. I. Numerical study of mass transfer in drop and film systems using a regularized finite difference scheme in evaporative lithography, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2018, vol. 22, no. 2, pp. 344–363 (In Russian). EDN: XWXSMX. DOI: https://doi.org/10.14498/vsgtu1601.
  29. Petsi A. J., Kalarakis A. N., Burganos V. N. Deposition of Brownian particles during evaporation of two-dimensional sessile droplets, Chem. Eng. Sci., 2010, vol. 65, no. 10, pp. 2978–2989. DOI: https://doi.org/10.1016/j.ces.2010.01.028.
  30. Hu S., Wang Y., Man X., Doi M. Deposition patterns of two neighboring droplets: Onsager variational principle studies, Langmuir, 2017, vol. 33, no. 23, pp. 5965–5972. DOI: https://doi.org/10.1021/acs.langmuir.7b01354.
  31. Yang J., Kim H., Lee C., et al. Phase-field modeling and computer simulation of the coffeering effect, Theor. Comput. Fluid Dyn., 2020, vol. 34, no. 5–6, pp. 679–692. DOI: https://doi.org/10.1007/s00162-020-00544-w.
  32. Seo H. W., Jung N., Yoo C. S. Oscillation dynamics of colloidal particles caused by surfactant in an evaporating droplet, J. Mech. Sci. Technol., 2020, vol. 34, no. 2, pp. 801–808. DOI: https://doi.org/10.1007/s12206-020-0128-1.
  33. Kim H.-S., Park S. S., Hagelberg F. Computational approach to drying a nanoparticlesuspended liquid droplet, J. Nanopart. Res., 2010, vol. 13, no. 1, pp. 59–68. DOI: https://doi.org/10.1007/s11051-010-0062-8.
  34. Crivoi A., Duan F. Three-dimensional Monte Carlo model of the coffee-ring effect in evaporating colloidal droplets, Sci. Rep., 2014, vol. 4, no. 1, 4310. DOI: https://doi.org/10.1038/srep04310.
  35. Zhang H., Shan Y. G., Li L., et al. Modeling the self-assembly of nanoparticles into branched aggregates from a sessile nanofluid droplet, Appl. Therm. Eng., 2016, vol. 94, pp. 650–656. DOI: https://doi.org/10.1016/j.applthermaleng.2015.10.160.
  36. Ren J., Crivoi A., Duan Fe. Disk-ring deposition in drying a sessile nanofluid droplet with enhanced Marangoni effect and particle surface adsorption, Langmuir, 2020, vol. 36, no. 49, pp. 15064–15074. DOI: https://doi.org/10.1021/acs.langmuir.0c02607.
  37. Ren J., Crivoi A., Duan F. Dendritic nanoparticle self-assembly from drying a sessile nanofluid droplet, Phys. Chem. Chem. Phys., 2021, vol. 23, no. 29, pp. 15774–15783. DOI: https://doi.org/10.1039/d1cp01181b.
  38. Lebedev-Stepanov P., Vlasov K. Simulation of self-assembly in an evaporating droplet of colloidal solution by dissipative particle dynamics, Colloids Surf. A Physicochem. Eng. Asp., 2013, vol. 432, pp. 132–138. EDN: RFHBFJ. DOI: https://doi.org/10.1016/j.colsurfa.2013.05.012.
  39. Breinlinger T., Kraft T. A simple method for simulating the coffee stain effect, Powder Technol., 2014, vol. 256, pp. 279–284. DOI: https://doi.org/10.1016/j.powtec.2014.02.024.
  40. Liu W., Midya J., Kappl M., et al. Segregation in drying binary colloidal droplets, ACS Nano, 2019, vol. 13, no. 5, pp. 4972–4979. DOI: https://doi.org/10.1021/acsnano.9b00459.
  41. Andac T., Weigmann P., Velu S. K. P., et al. Active matter alters the growth dynamics of coffee rings, Soft Matter, 2019, vol. 15, no. 7, pp. 1488–1496. DOI: https://doi.org/10.1039/c8sm01350k.
  42. Kolegov K. S. Monte Carlo simulation of colloidal particles dynamics in a drying drop, J. Phys.: Conf. Ser., 2019, vol. 1163, 012043. EDN: MSBXPO. DOI: https://doi.org/10.1088/1742-6596/1163/1/012043.
  43. Lebovka N. I., Tarasevich Yu. Yu., Bulavin L. A., et al. Sedimentation of a suspension of rods: Monte Carlo simulation of a continuous two-dimensional problem, Phys. Rev. E, 2019, vol. 99, no. 5, 052135. EDN: XNZLXM. DOI: https://doi.org/10.1103/physreve.99.052135.
  44. Kolegov K. S., Barash L. Yu. Joint effect of advection, diffusion, and capillary attraction on the spatial structure of particle depositions from evaporating droplets, Phys. Rev. E, 2019, vol. 100, no. 3, 033304. EDN: RUWORY. DOI: https://doi.org/10.1103/physreve.100.033304.
  45. Zolotarev P. A., Kolegov K. S. Average cluster size inside sediment left after droplet desiccation, J. Phys.: Conf. Ser., 2021, vol. 1740, 012029. EDN: OPMGGJ. DOI: https://doi.org/10.1088/1742-6596/1740/1/012029.
  46. Song R., Lee M., Moon H., et al. Particle dynamics in drying colloidal solution using discrete particle method, Flex. Print. Electron., 2021, vol. 6, no. 4, 044007. DOI: https://doi.org/10.1088/2058-8585/ac428e.
  47. Marinaro G., Riekel C., Gentile F. Size-exclusion particle separation driven by micro-flows in a quasi-spherical droplet: Modelling and experimental results, Micromachines, 2021, vol. 12, no. 2, 185. DOI: https://doi.org/10.3390/mi12020185.
  48. Hu H., Larson R. G. Analysis of the microfluid flow in an evaporating sessile droplet, Langmuir, 2005, vol. 21, no. 9, pp. 3963–3971. DOI: https://doi.org/10.1021/la047528s.
  49. Park Y., Park Y, Lee J., Lee C. Simulation for forming uniform inkjet-printed quantum dot layer, J. Appl. Phys., 2019, vol. 125, no. 6, 065304. DOI: https://doi.org/10.1063/1.5079863.
  50. Kolegov K. S. Simulation of mass transfer in droplet-film systems using a regularized difference scheme in evaporative lithography, Thesis of Dissertation (Cand. Phys. & Math. Sci.). Astrakhan, Astrakhan State University, 2018, 162 pp. (In Russian). DOI: https://doi.org/10.13140/RG.2.2.29663.97446
  51. Kolegov K. S. Simulation of patterned glass film formation in the evaporating colloidal liquid under IR heating, Microgravity Sci. Technol., 2018, vol. 30, no. 1–2, pp. 113–120. EDN: UXPYTS. DOI: https://doi.org/10.1007/s12217-017-9587-0.
  52. Bodiguel H., Leng J. Imaging the drying of a colloidal suspension, Soft Matter, 2010, vol. 6, pp. 5451–5460. DOI: https://doi.org/10.1039/C0SM00323A.
  53. Fischer B. J. Particle convection in an evaporating colloidal droplet, Langmuir, 2002, vol. 18, no. 1, pp. 60–67. DOI: https://doi.org/10.1021/la015518a.
  54. Dorodnitsyn L. V. Grid oscillations in finite-difference scheme and a method for their approximate analysis, Comput. Math. Model., 2016, vol. 27, no. 4, pp. 472–488. EDN: YUUJUV. DOI: https://doi.org/10.1007/s10598-016-9337-y.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. Figure 1. Scheme of the planned software complex for solving problems in the field of evaporative lithography

Download (283KB)
Indexing metadata
3. Figure 2. Numerical calculation algorithm

Download (229KB)
Indexing metadata
4. Figure 3. Comparison of the calculation results in two programs for several consecutive time points: (a) drop height, (b) mass fraction of dissolved or suspended matter, and (c) radial flow velocity averaged over the thickness of the liquid layer

Download (498KB)
Indexing metadata
5. Figure 4. The results of the calculation in Maple: the spatial-temporal dependence of (a) the radial flow velocity averaged over the thickness of the liquid layer and (b) the vapor flux density

Download (312KB)
Indexing metadata
6. Figure 5. Velocity vector field of fluid flow according results obtained with using the program written in C++ for the following times: (a) \(t=10\) s, (b) \(t=90\) s, (c) \(t=220\) s, and (d) \(t=300\) s

Download (371KB)
Indexing metadata
7. Figure 6. The results of the calculation in program written with C++: (a) capillary pressure and (b) possible sawtooth oscillations

Download (109KB)
Indexing metadata

Copyright (c) 2023 Authors; Samara State Technical University (Compilation, Design, and Layout)

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Согласие на обработку персональных данных с помощью сервиса «Яндекс.Метрика»

1. Я (далее – «Пользователь» или «Субъект персональных данных»), осуществляя использование сайта https://journals.rcsi.science/ (далее – «Сайт»), подтверждая свою полную дееспособность даю согласие на обработку персональных данных с использованием средств автоматизации Оператору - федеральному государственному бюджетному учреждению «Российский центр научной информации» (РЦНИ), далее – «Оператор», расположенному по адресу: 119991, г. Москва, Ленинский просп., д.32А, со следующими условиями.

2. Категории обрабатываемых данных: файлы «cookies» (куки-файлы). Файлы «cookie» – это небольшой текстовый файл, который веб-сервер может хранить в браузере Пользователя. Данные файлы веб-сервер загружает на устройство Пользователя при посещении им Сайта. При каждом следующем посещении Пользователем Сайта «cookie» файлы отправляются на Сайт Оператора. Данные файлы позволяют Сайту распознавать устройство Пользователя. Содержимое такого файла может как относиться, так и не относиться к персональным данным, в зависимости от того, содержит ли такой файл персональные данные или содержит обезличенные технические данные.

3. Цель обработки персональных данных: анализ пользовательской активности с помощью сервиса «Яндекс.Метрика».

4. Категории субъектов персональных данных: все Пользователи Сайта, которые дали согласие на обработку файлов «cookie».

5. Способы обработки: сбор, запись, систематизация, накопление, хранение, уточнение (обновление, изменение), извлечение, использование, передача (доступ, предоставление), блокирование, удаление, уничтожение персональных данных.

6. Срок обработки и хранения: до получения от Субъекта персональных данных требования о прекращении обработки/отзыва согласия.

7. Способ отзыва: заявление об отзыве в письменном виде путём его направления на адрес электронной почты Оператора: info@rcsi.science или путем письменного обращения по юридическому адресу: 119991, г. Москва, Ленинский просп., д.32А

8. Субъект персональных данных вправе запретить своему оборудованию прием этих данных или ограничить прием этих данных. При отказе от получения таких данных или при ограничении приема данных некоторые функции Сайта могут работать некорректно. Субъект персональных данных обязуется сам настроить свое оборудование таким способом, чтобы оно обеспечивало адекватный его желаниям режим работы и уровень защиты данных файлов «cookie», Оператор не предоставляет технологических и правовых консультаций на темы подобного характера.

9. Порядок уничтожения персональных данных при достижении цели их обработки или при наступлении иных законных оснований определяется Оператором в соответствии с законодательством Российской Федерации.

10. Я согласен/согласна квалифицировать в качестве своей простой электронной подписи под настоящим Согласием и под Политикой обработки персональных данных выполнение мною следующего действия на сайте: https://journals.rcsi.science/ нажатие мною на интерфейсе с текстом: «Сайт использует сервис «Яндекс.Метрика» (который использует файлы «cookie») на элемент с текстом «Принять и продолжить».