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

K. S. Kolegov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 27 (2023) no. 2, pp. 309-335.

Résumé

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.

Détail

Keywords: evaporating droplet, mass transfer, capillary flow, finite-difference scheme, sawtooth oscillation, suppression of the coffee-ring effect.

Bibliographie

[1] Zang D., Tarafdar S., Tarasevich Yu. Yu., et al., “Evaporation of a droplet: From physics to applications”, Phys. Rep., 804 (2019), 1–56 | DOI

[2] Kolegov K. S., Barash L. Yu., “Applying droplets and films in evaporative lithography”, Adv. Colloid Interf. Sci., 285 (2020), 102271 | DOI

[3] Deegan R. D., Bakajin O., Dupont T F., et al., “Capillary flow as the cause of ring stains from dried liquid drops”, Nature, 389:6653 (1997), 827–829 | DOI

[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., 36:3 (2022), 423–433 | DOI

[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., 50:5 (2021), 1011–1014 | DOI

[6] Wang W., Wang Q., Zhang K., et al., “On-demand contact line pinning during droplet evaporation”, Sens. Act. B: Chem., 312 (2020), 127983 | DOI

[7] Saroj S. K., Panigrahi P. K., “Magnetophoretic control of diamagnetic particles inside an evaporating droplet”, Langmuir, 37:51 (2021), 14950–14967 | DOI

[8] Al-Muzaiqer M. A., Kolegov K. S., Ivanova N. A., Fliagin V. M., “Nonuniform heating of a substrate in evaporative lithography”, Phys. Fluids, 33:9 (2021), 092101 | DOI

[9] Du F., Zhang L., Shen W., “Controllable dried patterns of colloidal drops”, J. Col. Int. Sci., 606 (2022), 758–767 | DOI

[10] Cedeno R., Grossier R., Lagaize M., et al., “Nucleation in sessile saline microdroplets: Induction time measurement via deliquescence-recrystallization cycling”, Faraday Discuss., 235 (2022), 183–197 | DOI

[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, 126:1 (2022), 505–516 | DOI

[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., 23:12 (2021), 7115–7124 | DOI

[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, 34:1 (2022), 017107 | DOI

[14] Kirner F., Sturm E. V., “Advances of nonclassical crystallization toward self-purification of precious metal nanoparticle mixtures”, Cryst. Growth Des., 21:9 (2021), 5192–5197 | DOI

[15] Hossain M. T., Gates I. D., Natale G., “Dynamics of Brownian Janus rods at a liquid-liquid interface”, Phys. Fluids, 34:1 (2022), 012117 | DOI

[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, 126:1 (2021), 327–335 | DOI

[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., 608 (2022), 873–881 | DOI

[18] Galy P. E., Guitton-Spassky T., Sella C., et al., “Redox control of particle deposition from drying drops”, ACS Appl. Mater. Interfaces, 14:2 (2022), 3374–3384 | DOI

[19] Inoue K., Inasawa S., “Drying-induced back flow of colloidal suspensions confined in thin unidirectional drying cells”, RSC Adv., 10:27 (2020), 15763–15768 | DOI

[20] Homede E., Manor O., “Deposition of nanoparticles from a volatile carrier liquid”, J. Col. Int. Sci., 562 (2020), 102–111 | DOI

[21] Li D., Chen R., Zhu X., et al., “Light-fueled beating coffee-ring deposition for droplet evaporative crystallization”, Anal. Chem., 93:25 (2021), 8817–8825 | DOI

[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., 128 (2021), 105573 | DOI

[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., 606 (2022), 2011–2023 | DOI

[24] Corletto A., Shapter J. G., “Thickness/morphology of functional material patterned by topographical discontinuous dewetting”, Nano Select, 2021, no. 9, 1723–1740 | DOI

[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, 6:2 (2020), e03382 | DOI

[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, 26:11 (2010), 7833–7842 | DOI

[27] Tarasevich Yu. Yu., Vodolazskaya I. V., Isakova O. P., “Desiccating colloidal sessile drop: Dynamics of shape and concentration”, Coll. Pol. Sci., 289:9 (2011), 1015–1023 | DOI

[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.], 22:2 (2018), 344–363 (In Russian) | DOI

[29] Petsi A. J., Kalarakis A. N., Burganos V. N., “Deposition of Brownian particles during evaporation of two-dimensional sessile droplets”, Chem. Eng. Sci., 65:10 (2010), 2978–2989 | DOI

[30] Hu S., Wang Y., Man X., Doi M., “Deposition patterns of two neighboring droplets: Onsager variational principle studies”, Langmuir, 33:23 (2017), 5965–5972 | DOI

[31] Yang J., Kim H., Lee C., et al., “Phase-field modeling and computer simulation of the coffee-ring effect”, Theor. Comput. Fluid Dyn., 34:5–6 (2020), 679–692 | DOI

[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., 34:2 (2020), 801–808 | DOI

[33] Kim H.-S., Park S. S., Hagelberg F., “Computational approach to drying a nanoparticle-suspended liquid droplet”, J. Nanopart. Res., 13:1 (2010), 59–68 | DOI

[34] Crivoi A., Duan F., “Three-dimensional Monte Carlo model of the coffee-ring effect in evaporating colloidal droplets”, Sci. Rep., 4:1 (2014), 4310 | DOI

[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., 94 (2016), 650–656 | DOI

[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, 36:49 (2020), 15064–15074 | DOI

[37] Ren J., Crivoi A., Duan F., “Dendritic nanoparticle self-assembly from drying a sessile nanofluid droplet”, Phys. Chem. Chem. Phys., 23:29 (2021), 15774–15783 | DOI

[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., 432 (2013), 132–138 | DOI

[39] Breinlinger T., Kraft T., “A simple method for simulating the coffee stain effect”, Powder Technol., 256 (2014), 279–284 | DOI

[40] Liu W., Midya J., Kappl M., et al., “Segregation in drying binary colloidal droplets”, ACS Nano, 13:5 (2019), 4972–4979 | DOI

[41] Andac T., Weigmann P., Velu S. K. P., et al., “Active matter alters the growth dynamics of coffee rings”, Soft Matter, 15:7 (2019), 1488–1496 | DOI

[42] Kolegov K. S., “Monte Carlo simulation of colloidal particles dynamics in a drying drop”, J. Phys.: Conf. Ser., 1163 (2019), 012043 | DOI

[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, 99:5 (2019), 052135 | DOI

[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, 100:3 (2019), 033304 | DOI

[45] Zolotarev P. A., Kolegov K. S., “Average cluster size inside sediment left after droplet desiccation”, J. Phys.: Conf. Ser., 1740 (2021), 012029 | DOI

[46] Song R., Lee M., Moon H., et al., “Particle dynamics in drying colloidal solution using discrete particle method”, Flex. Print. Electron., 6:4 (2021), 044007 | DOI

[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, 12:2 (2021), 185 | DOI

[48] Hu H., Larson R. G., “Analysis of the microfluid flow in an evaporating sessile droplet”, Langmuir, 21:9 (2005), 3963–3971 | DOI

[49] Park Y., Park Y, Lee J., Lee C., “Simulation for forming uniform inkjet-printed quantum dot layer”, J. Appl. Phys., 125:6 (2019), 065304 | DOI

[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 State University, Astrakhan, 2018, 162 pp. (In Russian) | DOI

[51] Kolegov K. S., “Simulation of patterned glass film formation in the evaporating colloidal liquid under IR heating”, Microgravity Sci. Technol., 30:1–2 (2018), 113–120 | DOI

[52] Bodiguel H., Leng J., “Imaging the drying of a colloidal suspension”, Soft Matter, 6 (2010), 5451–5460 | DOI

[53] Fischer B. J., “Particle convection in an evaporating colloidal droplet”, Langmuir, 18:1 (2002), 60–67 | DOI

[54] Dorodnitsyn L. V., “Grid oscillations in finite-difference scheme and a method for their approximate analysis”, Comput. Math. Model., 27:4 (2016), 472–488 | DOI | Zbl