Skip to main content
SpringerLink
Log in

Evaluation of the local kinetics of a colloidal system through diffusion–sedimentation phenomena: A  numerical approach

  • Original Contribution
  • Published:
Colloid and Polymer Science Aims and scope Submit manuscript

Abstract

This work presents a numerical solution to determine the concentration profiles of a colloidal system formed by titanium dioxide (TiO2) particles in water as a function of the time and the spatial coordinate through a classic mathematical model. The model describes the local kinetics given by the diffusion and sedimentation phenomena in the system. Using Python programming, local kinetics is solved numerically from partial differential equations (PDEs) using the finite difference method (FDM). For particle diameter > 200 nm, there are no significant changes in the concentration profiles, indicating then that the particle size does not directly influence the behavior of the system and thus maintains the law of mass conservation. The sedimentation time decreases, but the sedimentation time for particle diameter > 200 nm is practically the same (τ = 1.75). The numerical solution of the mathematical model has applications in production processes with industrial interest, particularly in the design and production of paints and coatings with long storage times.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Availability of data and materials

Not applicable.

References

  1. Catarino-Centeno R, Balderas-Altamirano MA, Pérez E, Gama-Goicochea A (2019) The role of solvent quality, inhomogeneous polymer brush composition, grafting density and number of free chains on the viscosity, friction between surfaces, and their scaling laws. Chem Phys Lett 722:124–131. https://doi.org/10.1016/j.cplett.2019.03.002

    Article  CAS  Google Scholar 

  2. Pasquale S, Zimbone M, Ruffino F, Stella G, Gueli AM (2021) Evaluation of the photocatalytic activity of water-based TiO2 nanoparticle dispersions applied on historical painting surfaces. Heritage 4(3):1854–1867. https://doi.org/10.3390/heritage4030104

    Article  Google Scholar 

  3. Shen X, Sun X, Liu J, Hang J, Jin L, Shi L (2021) A facile strategy to achieve monodispersity and stability of pigment TiO2 particles in low viscosity systems. J Colloid Interface Sci 581:586–594. https://doi.org/10.1016/j.jcis.2020.07.132

    Article  CAS  PubMed  Google Scholar 

  4. Dário BS, Pereira R, Petri DF (2022) Tristyrylphenol based surfactants as efficient dispersants of TiO2 particles in dilute and concentrated dispersions. Colloids and Surfaces A: Physicochem Eng Asp 654:130170. https://doi.org/10.1016/j.colsurfa.2022.130170

  5. Costa JR, Correia C, Góis JR, Silva SM, Antunes FE, Moniz J, Serra AC, Coelho JF (2017) Efficient dispersion of TiO2 using tailor made poly (acrylic acid)−based block copolymers, and its incorporation in water based paint formulation. Prog Org Coat 104:34–42. https://doi.org/10.1016/j.porgcoat.2016.12.006

    Article  CAS  Google Scholar 

  6. Calovi M, Coroneo V, Palanti S, Rossi S (2023) Colloidal silver as innovative multifunctional pigment: the effect of Ag concentration on the durability and biocidal activity of wood paints. Prog Org Coat 175:107354. https://doi.org/10.1016/j.porgcoat.2022.107354

  7. Gama-Goicochea A (2013) A model for the stability of a TiO2 dispersion. International Scholarly Research Notices Materials Science 2013:1–9. https://doi.org/10.1155/2013/547608

    Article  CAS  Google Scholar 

  8. Catarino-Centeno R, Pérez E, Gama-Goicochea A (2014) On the potential of mean force of a sterically stabilized dispersion. J Coat Technol Res 11(6):1023–1031. https://doi.org/10.1007/s11998-014-9600-0

    Article  CAS  Google Scholar 

  9. Bushell G (1998) Primary particle polydispersity in fractal aggregates. Dissertation, The University of New South Wales.

  10. Verwey EJW (1947) Theory of the stability of lyophobic colloids. J Phys Chem 51(3):631–636. https://doi.org/10.1021/j150453a001

    Article  CAS  Google Scholar 

  11. Liu W, Sun W, Borthwick AG, Ni J (2013) Comparison on aggregation and sedimentation of titanium dioxide, titanate nanotubes and titanate nanotubes-TiO2: influence of pH, ionic strength and natural organic matter. Colloids Surf, A 434:319–328. https://doi.org/10.1016/j.colsurfa.2013.05.010

    Article  CAS  Google Scholar 

  12. Karakaş F, Çelik MS (2013) Mechanism of TiO2 stabilization by low molecular weight NaPAA in reference to water-borne paint suspensions. Colloids Surf, A 434:185–193. https://doi.org/10.1016/j.colsurfa.2013.05.051

    Article  CAS  Google Scholar 

  13. Holmberg JP, Ahlberg E, Bergenholtz J, Hassellöv M, Abbas Z (2013) Surface charge and interfacial potential of titanium dioxide nanoparticles: experimental and theoretical investigations. J Colloid Interface Sci 407:168–176. https://doi.org/10.1016/j.jcis.2013.06.015

    Article  CAS  PubMed  Google Scholar 

  14. Vesaratchanon S, Nikolov A, Wasan DT (2007) Sedimentation in nano-colloidal dispersions: effects of collective interactions and particle charge. Adv Coll Interface Sci 134:268–278. https://doi.org/10.1016/j.cis.2007.04.026

    Article  CAS  Google Scholar 

  15. Al-Gebory LWI, Al-kaisy HA, Mahdi M (2020) Micro-hydrodynamic interaction mechanisms in tio2 nano-colloidal suspensions with different particle size distributions: the effects of electrostatic and steric stabilization. J Mech Eng Res Dev 43(2):226–238. https://jmerd.net/02-2020-226-238/

  16. Hsieh AH, Franses EI, Corti DS (2023) Effect of a double-chain surfactant on the stabilization of suspensions of silica and titania particles against agglomeration and sedimentation. Colloids Surf, A 662:130993

    Article  CAS  Google Scholar 

  17. Yang YJ, Kelkar AV, Corti DS, Franses EI (2016) Effect of interparticle interactions on agglomeration and sedimentation rates of colloidal silica microspheres. Langmuir 32(20):5111–5123. https://doi.org/10.1021/acs.langmuir.6b00925

    Article  CAS  PubMed  Google Scholar 

  18. Midelet J, El-Sagheer AH, Brown T, Kanaras AG, Werts MH (2017) The sedimentation of colloidal nanoparticles in solution and its study using quantitative digital photography. Part Part Syst Charact 34(10):1700095. https://doi.org/10.1002/ppsc.201700095

    Article  CAS  Google Scholar 

  19. Yang YJ, Kelkar AV, Zhu X, Bai G, Ng HT, Corti DS, Franses EI (2015) Effect of sodium dodecylsulfate monomers and micelles on the stability of aqueous dispersions of titanium dioxide pigment nanoparticles against agglomeration and sedimentation. J Colloid Interface Sci 450:434–445. https://doi.org/10.1016/j.jcis.2015.02.051

    Article  CAS  PubMed  Google Scholar 

  20. Cho EC, Zhang Q, Xia Y (2011) The effect of sedimentation and diffusion on cellular uptake of gold nanoparticles. Nat Nanotechnol 6(6):385–391. https://doi.org/10.1038/nnano.2011.58

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Cui J, Faria M, Björnmalm M et al (2016) A framework to account for sedimentation and diffusion in particle–cell interactions. Langmuir 32(47):12394–12402. https://doi.org/10.1021/acs.langmuir.6b01634

    Article  CAS  PubMed  Google Scholar 

  22. Mengual O, Meunier G, Cayré I, Puech K, Snabre P (1999) TURBISCAN MA 2000: multiple light scattering measurement for concentrated emulsion and suspension instability analysis. Talanta 50(2):445–456. https://doi.org/10.1016/S0039-9140(99)00129-0

    Article  CAS  PubMed  Google Scholar 

  23. Mason M, Weaver W (1924) The settling of small particles in a fluid. Phys Rev 23(3):412. https://doi.org/10.1103/PhysRev.23.412

    Article  CAS  Google Scholar 

  24. Giorgi F, Coglitore D, Curran JM et al (2019) The influence of inter-particle forces on diffusion at the nanoscale. Sci Rep 9(112689):1–6. https://doi.org/10.1038/s41598-019-48754-5

    Article  CAS  Google Scholar 

  25. Bermúdez B, Juárez L (2014) Solución Numérica de una Ecuación del Tipo Advección-Difusión. Información tecnológica 25(1):151–160. https://doi.org/10.4067/S0718-07642014000100016

    Article  Google Scholar 

  26. Yang J, Lee C, Kwak S, Choi Y, Kim J (2021) A conservative and stable explicit finite difference scheme for the diffusion equation. Journal of Computational Science 56:101–491. https://doi.org/10.1016/j.jocs.2021.101491

    Article  Google Scholar 

  27. Becerril R, Guzmán FS, Rendón-Romero A, Valdez-Alvarado S (2008) Solving the time–dependent Schrödinger. Revista mexicana de física E 54(2):120–132. https://rmf.smf.mx/ojs/index.php/rmf-e/article/view/4575/6088 Accessed 17 Feb 2023.

  28. Peza-Solís JF, Silva-Navarro G, Castro-Linares NR (2015) Trajectory tracking control in a single flexible-link robot using finite differences and sliding modes. Journal of applied research and technology 13(1):70–78. https://doi.org/10.1016/S1665-6423(15)30006-7

    Article  Google Scholar 

  29. Mannarino IA (2009) A mimetic finite difference method using crank–nicolson scheme for unsteady diffusion equation. Revista de matemática: Teoría y Aplicaciones 16(2):221–230. https://www.redalyc.org/pdf/453/45326951003.pdf Accessed 21 Jan 2023.

  30. Lange M, Kukreja N, Louboutin M, Luporini F, Vieira F, Pandolfo V, Velesko P, Kazakas P Gorman G (2016) Devito: towards a generic finite difference DSL using symbolic Python. 6th Workshop on Python for High-Performance and Scientific Computing (PyHPC) 67–75. https://doi.org/10.1109/PyHPC.2016.013

  31. Guyer JE, Wheeler D, Warren JA (2009) FiPy: partial differential equations with Python. Computing in Science & Engineering 11(3):6–15. https://doi.org/10.1109/MCSE.2009.52

    Article  CAS  Google Scholar 

  32. Cellier N, Ruyer-Quil C (2019) Scikit-finite-diff, a new tool for PDE solving. J Open Source Softw 4(38):1356. https://doi.org/10.21105/joss.01356

  33. Zwicker D (2020) py-pde: a Python package for solving partial differential equations. J Open Source Softw 5(48):2158. https://doi.org/10.21105/joss.02158

  34. Chang Q (2016) Colloid and interface chemistry for water quality control. Academic Press, China

    Google Scholar 

  35. Catarino-Centeno R, Waldo-Mendoza MA, García-Hernández E, Pérez-López JE (2021) Relationship between the coefficient of friction of additive in the bulk and chain graft surface density through a diffusion process: erucamide–stearyl erucamide mixtures in polypropylene films. J Vinyl Add Tech 27(2):459–466. https://doi.org/10.1002/vnl.21820

    Article  CAS  Google Scholar 

  36. Dhont JK (1996) An introduction to dynamics of colloids. Elsevier, The Netherlands

    Google Scholar 

  37. Mariño-Pérez A, Falcón-Hernández J, Valadao GES (2010) Correlación teórica entre las concentraciones de sólidos en el lodo sedimentado por gravedad y en la la torta sin escurrir. Minería y Geología 26(4):79–103. https://www.redalyc.org/pdf/2235/223515971004.pdf Accessed: 23 Feb 2023.

  38. Treiman S, Jackiw R (2014) Current algebra and anomalies. Princeton University Press, United States, In Current Algebra and Anomalies

    Book  Google Scholar 

  39. Hirsch C (2007) Numerical computation of internal and external flows: the fundamentals of computational fluid dynamics. Elsevier, Great Britain

    Google Scholar 

  40. Cortés-Rosas JJ, González-Cárdenas, M. E., Pinilla-Morán, VD, Salazar-Moreno A, Tovar-Pérez VH (2019) Solución numérica de ecuaciones diferenciales parciales. Technical report, UNAM. https://www.ingenieria.unam.mx/pinilla/PE105117/pdfs/tema6/6-1-2-3_edparciales.pdf. Accessed 31 Jan 2023

  41. Logan JD (1998) The physical origins of partial differential equations. In Applied Partial Differential Equations. Springer, New York

  42. Logan JD (1998) Applied partial differential equations. Springer-Verlag, New York

    Book  Google Scholar 

  43. Chapra SC, Canale RR, Ruiz RS, Mercado VHI, Díaz EM, Benites GE (2015) Métodos numéricos para ingenieros. McGraw-Hill, México

    Google Scholar 

  44. Hildebrand FB (1956) Introduction to numerical analysis. McGraw-Hill, New York

    Google Scholar 

  45. Pletcher RH, Tannehill JC, Anderson DA (2013) Computational fluid mechanics and heat transfer. CRC Press, New York

    Google Scholar 

  46. Python website. https://www.python.org/ Accessed 12 Dec 2022

  47. Challenger-Pérez I, Díaz-Ricardo Y, Becerra-García RA (2014) El lenguaje de programación Python, Ciencias Holguín. https://www.redalyc.org/articulo.oa?id=181531232001. Accessed 18 Oct 2022

  48. Boinovich LB (2007) Long-range surface forces and their role in the progress. Russ Chem Rev 76(5):471–488. https://doi.org/10.1070/RC2007v076n05ABEH003692

    Article  CAS  Google Scholar 

  49. Liu X, Chen G, Su C (2011) Effects of material properties on sedimentation and aggregation of titanium dioxide nanoparticles of anatase and rutile in the aqueous phase. J Colloid Interface Sci 363(1):84–91. https://doi.org/10.1016/j.jcis.2011.06.085

    Article  CAS  PubMed  Google Scholar 

  50. Ganguly S, Chakraborty S (2011) Sedimentation of nanoparticles in nanoscale colloidal suspensions. Phys Lett A 375(24):2394–2399. https://doi.org/10.1016/j.physleta.2011.04.018

    Article  CAS  Google Scholar 

Download references

Acknowledgements

RCC and EGH are grateful for the computer resources and support provided by LANCAD at the High-Performance Cluster Yoltla at UAM Iztapalapa.

Funding

This work was partially supported by the Tecnológico Nacional de México.

Author information

Authors and Affiliations

  1. Tecnológico Nacional de México/Instituto Tecnológico Superior de Zacapoaxtla, División de Ingeniería Mecatrónica, Zacapoaxtla, Zacapoaxtla, Puebla, 73680, Mexico

    Rafael Catarino-Centeno, Gabriela Hilario-Acuapan & Erwin García-Hernández

  2. Instituto Tecnológico de Celaya/Tecnológico Nacional de México, Departamento de Ingeniería Química, Celaya, Guanajuato, 38010, Mexico

    Rosalba Patiño-Herrera

Authors
  1. Rafael Catarino-Centeno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriela Hilario-Acuapan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rosalba Patiño-Herrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Erwin García-Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

R.C.C. conceptualization; methodology; investigation; formal analysis; writing, original draft; writing, review and editing; supervision; and funding acquisition. G.H.A. formal analysis and writing, review and editing. R.P.H. review and editing. E.G.H. writing, review, and editing. All authors reviewed the manuscript.

Corresponding author

Correspondence to Rafael Catarino-Centeno.

Ethics declarations

Ethical approval

Not applicable.

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Catarino-Centeno, R., Hilario-Acuapan, G., Patiño-Herrera, R. et al. Evaluation of the local kinetics of a colloidal system through diffusion–sedimentation phenomena: A  numerical approach. Colloid Polym Sci 301, 1437–1448 (2023). https://doi.org/10.1007/s00396-023-05160-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00396-023-05160-8

Keywords

Search

Navigation