Advertisement

Kinetics simulation of transmembrane transport of ions and molecules through a semipermeable membrane

  • S. O. KarakhimEmail author
  • P. F. Zhuk
  • S. O. Kosterin
Article
  • 25 Downloads

Abstract

We have developed a model to study the kinetics of the redistribution of ions and molecules through a semipermeable membrane in complex mixtures of substances penetrating and nonpenetrating through a membrane. It takes into account the degree of dissociation of these substances, their initial concentrations in solutions separated by a membrane, and volumes of these solutions. The model is based on the assumption that only uncharged particles (molecules or ion pairs) diffuse through a membrane (and not ions as in the Donnan model). The developed model makes it possible to calculate the temporal dependencies of concentrations for all processing ions and molecules at system transition from the initial state to equilibrium. Under equilibrium conditions, the ratio of ion concentrations in solutions separated by a membrane obeys the Donnan distribution. The Donnan effect is the result of three factors: equality of equilibrium concentrations of penetrating molecules on each side of a membrane, dissociation of molecules into ions, and Le Chatelier’s principle. It is shown that the Donnan distribution (irregularity of ion distribution) and accordingly absolute value of the Donnan membrane potential increases if: (i) the nonpenetrating salt concentration (in one of the solutions) and its dissociation constant increases, (ii) the total penetrating salt concentration and its dissociation constant decreases, and (iii) the volumes ratio increases (between solutions with and without a nonpenetrating substance). It is shown also that only a slight difference between the degrees of dissociation of two substances can be used for their membrane separation.

Keywords

Membrane transport Kinetic model Membrane permeability Donnan distribution Donnan potential 

Notes

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

  1. Al-Obaidi MA, Kara-Zaitri C, Mujtaba IM (2017) Scope and limitations of the irreversible thermodynamics and the solution diffusion models for the separation of binary and multi-component systems in reverse osmosis process. Comput Chem Eng 100:48–79.  https://doi.org/10.1016/j.compchemeng.2017.02.001 CrossRefGoogle Scholar
  2. Cohen H, Cooley JW (1965) The numerical solution of the time-dependent Nernst-Planck equations. Biophys J 5:145–162.  https://doi.org/10.1016/S0006-3495(65)86707-8 CrossRefPubMedPubMedCentralGoogle Scholar
  3. Davis TA (2000) Donnan dialysis. In: Wilson ID, Adlard ER, Cooke M, Poole CF (eds) Encyclopedia of separation science. Academic Press, London, pp 1701–1707CrossRefGoogle Scholar
  4. Déon S, Escoda A, Fievet P, Salut R (2013) Prediction of single salt rejection by NF membranes: an experimental methodology to assess physical parameters from membrane and streaming potentials. Desalination 315:37–45.  https://doi.org/10.1016/j.desal.2012.09.005 CrossRefGoogle Scholar
  5. Donnan FG (1924) The theory of membrane equilibria. Chem Rev 1:73–90.  https://doi.org/10.1021/cr60001a003 CrossRefGoogle Scholar
  6. Donnan FG (1995) Theory of membrane equilibria and membrane potentials in the presence of non-dialysing electrolytes. A contribution to physical-chemical physiology. J Membr Sci 100:45–55.  https://doi.org/10.1016/0376-7388(94)00297-C CrossRefGoogle Scholar
  7. Duffey ME, Fennell Evans D, Cussler EL (1978) Simultaneous diffusion of ions and ion pairs across liquid membranes. J Membr Sci 3:1–14.  https://doi.org/10.1016/S0376-7388(00)80407-X CrossRefGoogle Scholar
  8. Fridman-Bishop N, Tankus KA, Freger V (2018) Permeation mechanism and interplay between ions in nanofiltration. J Membr Sci 548:449–458.  https://doi.org/10.1016/j.memsci.2017.11.050 CrossRefGoogle Scholar
  9. Galach M, Waniewski J (2012) Membrane transport of several ions during peritoneal dialysis: mathematical modeling. Artif Organs 36:E163–E178.  https://doi.org/10.1111/j.1525-1594.2012.01484.x CrossRefPubMedGoogle Scholar
  10. Galama AH, Post JW, Hamelers HVM, Nikonenko VV, Biesheuvel PM (2016) On the origin of the membrane potential arising across densely charged ion exchange membranes: how well does the Teorell-Meyer-Sievers theory work? J Membr Sci Res 2:128–140.  https://doi.org/10.22079/jmsr.2016.20311 CrossRefGoogle Scholar
  11. Gimmi T, Alt-Epping P (2018) Simulating Donnan equilibria based on the Nernst-Planck equation. Geochim Cosmochim Acta 232:1–13.  https://doi.org/10.1016/j.gca.2018.04.003 CrossRefGoogle Scholar
  12. Grzegorczyn S, Ślęzak A (2006) Time characteristics of electromotive force in single-membrane cell for stable and unstable conditions of reconstructing of concentration boundary layers. J Membr Sci 280:485–493.  https://doi.org/10.1016/j.memsci.2006.02.004 CrossRefGoogle Scholar
  13. Higa M, Kira A (1992) Theory and simulation of ion transport in nonstationary states against concentration gradients across ion-exchange membranes. J Phys Chem 96:9518–9523.  https://doi.org/10.1021/j100202a081 CrossRefGoogle Scholar
  14. Johnson KS, Pytkowicz RM (1978) Ion association of Cl with H+, Na+, K+, Ca2+, and Mg2+ in aqueous solutions at 25° C. Am J Sci 278:1428–1447.  https://doi.org/10.2475/ajs.278.10.1428 CrossRefGoogle Scholar
  15. Kim DY, Lee MH, Boram G, Kim JH, Lee S, Yang DR (2010) Modeling of solute transport in multi-component solution for reverse osmosis membranes. Desalination Water Treat 15:20–28.  https://doi.org/10.5004/dwt.2010.1662 CrossRefGoogle Scholar
  16. Kondepudi D, Prigogine I (1998) Modern thermodynamics. From heat engines to dissipative structures. John Wiley & Sons, New YorkGoogle Scholar
  17. Kosterin SA, Cherny AP (1991) Gibbs-Donnan equilibrium in the system membrane vesicules – incubation medium. Biofizika 36:826–829. (In Russian)Google Scholar
  18. Kozmai A, Chérif M, Dammak L, Bdiri M, Larchet C, Nikonenko V (2017) Modelling non-stationary ion transfer in neutralization dialysis. J Membr Sci 540:60–70.  https://doi.org/10.1016/j.memsci.2017.06.039 CrossRefGoogle Scholar
  19. Kumaran M, Bajpai S (2015) Application of extended Nernst Planck model in nano filtration process –a critical review. Int J Eng Res Rev 3:40–49Google Scholar
  20. Kurbel S (2011) Donnan effect on chloride ion distribution as a determinant of body fluid composition that allows action potentials to spread via fast sodium channels. Theor Biol Med Model 8:16.  https://doi.org/10.1186/1742-4682-8-16 CrossRefPubMedPubMedCentralGoogle Scholar
  21. Lang GE, Stewart PS, Vella D, Waters SL, Goriely A (2014) Is the Donnan effect sufficient to explain swelling in brain tissue slices? J Roy Soc Interface 11:20140123.  https://doi.org/10.1098/rsif.2014.0123 CrossRefGoogle Scholar
  22. Luo J, Wu C, Wu Y, Xu T (2013) Diffusion dialysis of hydrochloric acid with their salts: effect of co-existence metal ions. Sep Purif Technol 118:716–722.  https://doi.org/10.1016/j.seppur.2013.08.014 CrossRefGoogle Scholar
  23. Marcus Y, Hefter G (2006) Ion pairing. Chem Rev 106:4585–4621.  https://doi.org/10.1021/cr040087x CrossRefPubMedGoogle Scholar
  24. Mazur I, Kosterin S, Veklich T, Shkrabak O (2014) Gibbs-Donnan potential as a tool for membrane vesicles polarization. J Biophys Chem 5:78–89.  https://doi.org/10.4236/jbpc.2014.52009 CrossRefGoogle Scholar
  25. Moshtarikhah S, Oppers NAW, de Groot MT, Keurentjes JTF, Schouten JC, van der Schaaf J (2017) Nernst-Planck modeling of multicomponent ion transport in a Naflon membrane at high current density. J Appl Electrochem 47:51–62.  https://doi.org/10.1007/s10800-016-1017-2 CrossRefGoogle Scholar
  26. Neihof R, Sollner K (1957) The transitory overshooting of final equilibrium concentrations in membrane systems which drift toward the Gibbs-Donnan membrane equilibrium. J Phys Chem 61:159–163.  https://doi.org/10.1021/j150548a008 CrossRefGoogle Scholar
  27. Nguyen MK, Kurtz I (2006) Quantitative interrelationship between Gibbs-Donnan equilibrium, osmolality of body fluid compartments, and plasma water sodium concentration. J Appl Physiol 100:1293–1300.  https://doi.org/10.1152/japplphysiol.01274.2005 CrossRefPubMedGoogle Scholar
  28. Nouri S, Dammak L, Bulvestre G, Auclair B (2002) Studies of the crossed ionic fluxes through a cation-exchange membrane in the case of Donnan dialysis. Desalination 148:383–388.  https://doi.org/10.1016/S0011-9164(02)00734-8 CrossRefGoogle Scholar
  29. Osterhout WJV (1925) Is living protoplasm permeable to ions? J Gen Physiol 8:131–146.  https://doi.org/10.1085/jgp.8.2.131 CrossRefPubMedPubMedCentralGoogle Scholar
  30. Osterhout WJV (1929) The kinetics of penetration. J Gen Physiol 13:261–294.  https://doi.org/10.1085/jgp.13.2.261 CrossRefGoogle Scholar
  31. Palmeri J, Lefebvre X (2006) Computer simulation of Nanofiltration, membranes and processes. In: Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology, Transport Phenomena and Nanoscale Processes, vol 5, 1st edn. American Scientific Publishers, Stevenson Ranch, pp 93–214Google Scholar
  32. Philipse A, Vrij A (2011) The Donnan equilibrium: I. on the thermodynamic foundation of the Donnan equation of state. J Phys Condens Matter 23:194106.  https://doi.org/10.1088/0953-8984/23/19/194106 CrossRefPubMedGoogle Scholar
  33. Prado-Rubio OA, Møllerhøj M, Jørgensen SB, Jonsson G (2010) Modeling Donnan dialysis separation for carboxylic anion recovery. Comput Chem Eng 34:1567–1579.  https://doi.org/10.1016/j.compchemeng.2010.03.003 CrossRefGoogle Scholar
  34. Pyrzynska K (2006) Preconcentration and recovery of metal ions by Donnan dialysis. Microchim Acta 153:117–126.  https://doi.org/10.1007/s00604-005-0434-4 CrossRefGoogle Scholar
  35. Ramirez P, Alcaraz A, Mafe S, Pellicer J (2002) Donnan equilibrium of ionic drugs in pH-dependent fixed charge membranes: theoretical modelling. J Colloid Interface Sci 253:171–179.  https://doi.org/10.1006/jcis.2002.8508 CrossRefPubMedGoogle Scholar
  36. Rohman FS, Aziz N (2008) Mathematical model of ion transport in electrodialysis process. Bull Chem React Eng Catal 3:3–8.  https://doi.org/10.9767/bcrec.3.1-3.7122.3-8 CrossRefGoogle Scholar
  37. Sarkar S, Sengupta A, Prakash P (2010) The Donnan membrane principle: opportunities for sustainable engineered processes and materials. Environ Sci Technol 44:1161–1166.  https://doi.org/10.1021/es9024029 CrossRefPubMedGoogle Scholar
  38. Shu L, Liu X, Li Y, Yang B, Huang S, Lin Y, Jin S (2016) Modified Kedem-Katchalsky equations for osmosis through nano-pore. Desalination 399:47–52.  https://doi.org/10.1016/j.desal.2016.08.011 CrossRefGoogle Scholar
  39. Sobana S, Panda RC (2011) Review on modelling and control of desalination system using reverse osmosis. Rev Environ Sci Biotechnol 10:139–150.  https://doi.org/10.1007/s11157-011-9233-z CrossRefGoogle Scholar
  40. Steele A, Arias J (2014) Accounting for the Donnan effect in diafiltration optimization for high concentration UFDF applications. BioProcess Int 12:50–54Google Scholar
  41. Szczepański P, Szczepańska G (2017) Donnan dialysis − a new predictive model for non−steady state transport. J Membr Sci 525:277–289.  https://doi.org/10.1016/j.memsci.2016.11.017 CrossRefGoogle Scholar
  42. Tanaka Y (2012) Measurement of membrane characteristics using the phenomenological equation and the overall mass transport equation in ion-exchange membrane electrodialysis of saline water. Int J Chem Eng 2012:Article ID 148147, 12.  https://doi.org/10.1155/2012/148147 CrossRefGoogle Scholar
  43. Tian H, Zhang L, Wang M (2015) Applicability of Donnan equilibrium theory at nanochannel-reservoir interfaces. J Colloid Interface Sci 452:78–88.  https://doi.org/10.1016/j.jcis.2015.03.064 CrossRefPubMedGoogle Scholar
  44. Vega FA, Weng L, Temminghoff EJM, Van Riemsdijk WH (2010) Donnan membrane technique (DMT) for anion measurement. Anal Chem 82:2932–2939.  https://doi.org/10.1021/ac9029339 CrossRefPubMedGoogle Scholar
  45. Volpert AI, Hudyaev SI (1975) Analyses in classes of discontinuous functions and equations of mathematical physics. Nauka, Moscow. (In Russian)Google Scholar
  46. Wang J, Dlamini DS, Mishra AK, Pendergast MTM, Wong MCY, Mamba BB, Freger V, Verliefde ARD, Hoek EMV (2014) A critical review of transport through osmotic membranes. J Membr Sci 454:516–537.  https://doi.org/10.1016/j.memsci.2013.12.034 CrossRefGoogle Scholar
  47. Yaroshchuk A, Martínez-Lladó X, Llenas L, Rovira M, de Pablo J (2011) Solution-diffusion-film model for the description of pressure-driven trans-membrane transfer of electrolyte mixtures: one dominant salt and trace ions. J Membr Sci 368:192–201.  https://doi.org/10.1016/j.memsci.2010.11.037 CrossRefGoogle Scholar
  48. Zhao R, Van Soestbergen M, Rijnaarts HHM, Van der Wal A, Bazant MZ, Biesheuvel PM (2012) Time-dependent ion selectivity in capacitive charging of porous electrodes. J Colloid Interface Sci 384:38–44.  https://doi.org/10.1016/j.jcis.2012.06.022 CrossRefPubMedGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Palladin Institute of Biochemistry of the National Academy of Sciences of UkraineKyivUkraine
  2. 2.National Aviation UniversityKyivUkraine

Personalised recommendations