Skip to main content
SpringerLink
Log in

A predictive procedure to model gas transport and intrinsic properties of rubbery polymeric membranes using equilibrium thermodynamics and free volume theory

  • Original Paper
  • Published:
Journal of Polymer Research Aims and scope Submit manuscript

Abstract

In this study, the lattice fluid (LF) and Vrentas’s self-diffusion (VSD) model were employed to predict the gas sorption, diffusion, and permeability through various rubbery polymeric membranes, like PDMS, PB, PE, and PIP as homopolymers and SBR (36 wt.%) and EVAc (54 wt.%) as copolymers. The LF model with use of variable characteristic parameters that were based on the temperature and pressure range of interest, enabled to predict convex behavior of pressure-induced sorption of PDMS and PB with an AARD < 3%. The Dullien method was also used to calculate the self-diffusion coefficient of pure components in the selected polymers. Extending VSD model led to reliable gas diffusion predictions with AARD < 10% for all systems. Furthermore, using concept of bulk modulus in combination with Tait equation, pressure functionality of diffusion factor was successfully corrected. The pressure and temperature dependency of the gas permeability through the selected homopolymers and copolymers were also investigated by the proposed model, and it was observed that the predicted permeability values were consistent with the experimental data.

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
Fig. 8

Similar content being viewed by others

Data availability

The authors confirm that the data supporting the findings of this study are available within the article.

Abbreviations

P i :

Permeability of component i (barrer)

L :

Membrane thickness (μm)

∆p :

Pressure difference of up and downstream of membrane (bar)

x :

Axis of x-direction (-)

D i :

Self-diffusion coefficient of component i

R :

Universal gas constant (j.mol−1.K−1)

T :

Absolute temperature (K)

N p :

Number of penetrants (-)

w i :

Mass fraction of component i

r i :

Molecular length of component i

\(\widetilde{p}\) :

Reduced pressure of mixture (-)

\(\widetilde{T}\) :

Reduced temperature of mixture (-)

T * i :

Characteristic temperature of component i (K)

p * i :

Characteristic pressure of component i (MPa)

∆p * ij :

Interaction parameter of LF model with respect to i and j component (MPa)

r :

Molecular length of mixture (-)

M i :

Molecular mass of component i (g.mol−1)

x i :

Mole fraction of component i

k :

Slope of bulk modulus function versus pressure in the rubbery state (-)

K * :

Bulk modulus of polymer in the rubbery state at reference temperature (bar)

k c :

Slope of bulk modulus function versus pressure in the glassy state (-)

K * c :

Bulk modulus of polymer in the glassy state at reference temperature (bar)

\({\widehat{V}}_{FH.i}\) :

Hole free-volume of component i (cm3.g−1)

\({\widehat{V}}_{iJ}\) :

Volume of molecular jumping unit of component i (cm3.g−1)

D oi :

Pre-exponential factor of VSD model (cm2.s−1)

E :

Activation energy of VSD model (j.mol−1)

K I,i :

Slope of hole free-volume relation versus temperature (cm3.mol−1.K−1)

K II,i :

Second parameter of hole free-volume relation (K)

T g,i :

Glass transition temperature of component i (K)

\(\widehat{A}/\widehat{B}\) :

Aspect ratio of gaseous penetrants (-)

a, b, a ’ and b ’ :

Parameters of Hariharan relation for each characteristic parameter

X * :

Represents for each characteristic parameter in the Boudouris relation

N i :

Number of first order parameters of i-th group in the repeating unit

C i :

First order parameters of i-th group in the repeating unit

M j :

Number of second order parameters of i-th group in the repeating unit

D j :

Second order parameters of i-th group in the repeating unit

V c,i :

Critical volume of component i (cm3.mol.−1)

V i :

Molar volume of component i (cm3.mol.−1)

E (w i \(\to\) 1) :

Activation energy in the limit of zero mass fraction of polymer

AARD % :

Absolute average relative deviation (%)

φ i :

Volume fraction of component i

φ i Eq . :

Volume fraction of component i at equilibrium state

ρ i :

Density of pure component i (g.cm−3)

μ i :

Chemical potential of component i (j.mol−1)

\(\widetilde{\rho }\) :

Reduced density of mixture (-)

ρ * i :

Characteristic density of component i (g.cm−3)

υ * i :

Characteristic volume of component i (cm3.mol−1)

υ * :

Characteristic volume of mixture (cm3.mol−1)

δ ij :

Interaction parameter of LF model (-)

α :

Thermal expansion of polymer in the rubbery state (K−1)

κ :

Compressibility factor of polymer in the rubbery state (bar−1)

γ :

Gas induced plasticization factor of polymer in the rubbery state (-)

γ i :

Overlap factor of component i (-)

θ i :

Pressure correction of free volume relation (-)

ξ ij :

Interaction parameter of VSD model (-)

i :

Pure gas component i

j :

Pure gas component j

p :

Pure polymer component p

\(\sum\) :

All components except the n-th one

c :

Summation of occupied and interstitial volume

References

  1. Iulianelli A, Drioli E (2020) Membrane engineering: Latest advancements in gas separation and pre-treatment processes, petrochemical industry and refinery, and future perspectives in emerging applications. Fuel Process Technol 206:106464. https://doi.org/10.1016/j.fuproc.2020.106464

    Article  CAS  Google Scholar 

  2. Mukhtar A, Saqib S, Mellon NB, Babar M, Rafiq S, Ullah S, Bustam MA, Al-Sehemi AG, Muhammad N, Chawla M (2020) CO2 capturing, thermo-kinetic principles, synthesis and amine functionalization of covalent organic polymers for CO2 separation from natural gas: A review. J. Nat. Gas Sci. Eng. 77:103203. https://doi.org/10.1016/j.jngse.2020.103203

    Article  CAS  Google Scholar 

  3. Zakaria Z, Shaari N, Kamarudin SK, Bahru R, Musa MT (2020) A review of progressive advanced polymer nanohybrid membrane in fuel cell application. Int J Energy Res 44:8255–8295. https://doi.org/10.1002/er.5516

    Article  CAS  Google Scholar 

  4. Liu G, Cheng L, Chen G, Liang F, Liu G, Jin W (2020) Pebax-Based Membrane Filled with Two-Dimensional Mxene Nanosheets for Efficient CO2 Capture. Chem - An Asian J 15:2364–2370. https://doi.org/10.1002/asia.201901433

    Article  CAS  Google Scholar 

  5. Selyanchyn O, Selyanchyn R, Fujikawa S (2020) Critical Role of the Molecular Interface in Double-Layered Pebax-1657/PDMS Nanomembranes for Highly Efficient CO2/N2Gas Separation. ACS Appl Mater Interfaces 12:33196–33209. https://doi.org/10.1021/acsami.0c07344

    Article  CAS  PubMed  Google Scholar 

  6. Daglar H, Erucar I, Keskin S (2021) Exploring the performance limits of MOF/polymer MMMs for O2/N2 separation using computational screening. J. Memb. Sci. 618:118555. https://doi.org/10.1016/j.memsci.2020.118555

    Article  CAS  Google Scholar 

  7. Huang Y, Xiao C, Huang Q, Liu H, Zhao J (2021) Progress on polymeric hollow fiber membrane preparation technique from the perspective of green and sustainable development. Chem Eng J 403:126295. https://doi.org/10.1016/j.cej.2020.126295

    Article  CAS  Google Scholar 

  8. Baker RW (2000) Membrane technology and applications. First, McGraw-Hill

  9. Yampolskii Y, Pinnau I, Freeman B (2006) Transport of gases and vapors in glassy and rubbery polymers. In: P.I. and F.B. Yampolskii Y (ed), Mater. Sci. Membr. Gas Vap. Sep., John Wiley & Sons Ltd, West Sussex, United Kingdom, pp 1–47

  10. Genduso G, Litwiller E, Ma X, Zampini S, Pinnau I (2019) Mixed-gas sorption in polymers via a new barometric test system: sorption and diffusion of CO2-CH4 mixtures in polydimethylsiloxane (PDMS). J Memb Sci 577:195–204. https://doi.org/10.1016/j.memsci.2019.01.046

    Article  CAS  Google Scholar 

  11. Merkel TC, Bondar VI, Nagai K, Freeman BD, Pinnau I (2000) Gas sorption, diffusion, and permeation in poly(dimethylsiloxane). J Polym Sci Part B Polym Phys 38:415–434. https://doi.org/10.1002/(SICI)1099-0488(20000201)38:3%3c415::AID-POLB8%3e3.0.CO;2-Z

    Article  CAS  Google Scholar 

  12. Xing R, Ho WSW (2009) Synthesis and characterization of crosslinked polyvinylalcohol/polyethyleneglycol blend membranes for CO2/CH4 separation. J Taiwan Inst Chem Eng 40:654–662. https://doi.org/10.1016/j.jtice.2009.05.004

    Article  CAS  Google Scholar 

  13. Dutta RC, Bhatia SK (2017) Transport diffusion of light gases in polyethylene using atomistic simulations. Langmuir 33:936–946. https://doi.org/10.1021/acs.langmuir.6b04037

    Article  CAS  PubMed  Google Scholar 

  14. Kamiya Y, Naito Y, Bourbon D (1989) Sorption and Partial Molar Volume of Gases in Polybutadiene. J Polym Sci Part B Polym Phys 27:2243–2250. https://doi.org/10.1002/polb.1989.090271107

    Article  CAS  Google Scholar 

  15. Lin H, Freeman BD (2004) Gas solubility, diffusivity and permeability in poly(ethylene oxide). J Memb Sci 239:105–117. https://doi.org/10.1016/j.memsci.2003.08.031

    Article  CAS  Google Scholar 

  16. Minelli M, Doghieri F (2017) Predictive model for gas and vapor sorption and swelling in glassy polymers: II. Effect of sample previous history. Fluid Phase Equilib 444:47–55. https://doi.org/10.1016/j.fluid.2017.04.012

    Article  CAS  Google Scholar 

  17. Vrentas JS, Vrentas CM (1998) Predictive methods for self-diffusion and mutual diffusion coefficients in polymer-solvent systems. Eur Polym J 34:797–803. https://doi.org/10.1016/S0014-3057(97)00205-X

    Article  CAS  Google Scholar 

  18. Nabati S, Raisi A, Aroujalian A (2017) Modeling of gas solubility and permeability in glassy and rubbery membranes using lattice fluid theory. Polymer (Guildf) 115:184–196. https://doi.org/10.1016/j.polymer.2017.03.032

    Article  CAS  Google Scholar 

  19. Sanchez IC, Lacombe RH (1978) Statistical Thermodynamics of Polymer Solutions. Macromolecules 11:1145–1156. https://doi.org/10.1021/ma60066a017

    Article  CAS  Google Scholar 

  20. Shoghl SN, Raisi A, Aroujalian A (2016) A predictive model for gas and vapor sorption into glassy membranes at high pressure. RSC Adv 6:57683–57694. https://doi.org/10.1039/c6ra07331j

    Article  CAS  Google Scholar 

  21. Flory PJ, Orwoll RA, Vrij A (1964) Statistical thermodynamics of chain molecule liquids. II. Liquid mixtures of normal paraffin hydrocarbons. J Am Chem Soc 86:3515–3520. https://doi.org/10.1021/ja01071a024

    Article  CAS  Google Scholar 

  22. Hariharan R, Freeman BD, Carbonell RG, Sarti GC (1993) Equation of state predictions of sorption isotherms in polymeric materials. J Appl Polym Sci 50:1781–1795. https://doi.org/10.1002/app.1993.070501014

    Article  CAS  Google Scholar 

  23. Davis PK, Lundy GD, Palamara JE, Duda JL, Danner RP (2004) Diffusion in polymers at elevated pressures. Ind Eng Chem Res. https://doi.org/10.1021/ie034075y

    Article  Google Scholar 

  24. Boudouris D, Constantinou L, Panayiotou C (2000) Prediction of volumetric behavior and glass transition temperature of polymers: A group contribution approach. Fluid Phase Equilib 167:1–19. https://doi.org/10.1016/S0378-3812(99)00286-1

    Article  CAS  Google Scholar 

  25. Doolittle AK (1951) Studies in Newtonian flow. I. The dependence of the viscosity of liquids on temperature. J Appl Phys 22:1031–1035. https://doi.org/10.1063/1.1700096

    Article  CAS  Google Scholar 

  26. Vrentas JS, Duda JL (1977) Diffusion in polymer–solvent systems. II. A predictive theory for the dependence of diffusion coefficients on temperature, concentration, and molecular weight. J Polym Sci Polym Phys Ed 15:417–439. https://doi.org/10.1002/pol.1977.180150303

    Article  CAS  Google Scholar 

  27. Thornton AW, Nairn KM, Hill AJ, Hill JM, Huang Y (2009) New relation between diffusion and free volume: II. Predicting vacancy diffusion. J. Memb. Sci. 338:38–42. https://doi.org/10.1016/j.memsci.2009.03.057

    Article  CAS  Google Scholar 

  28. Faridi N, Hadj-Romdhane I, Danner RP, Duda JL (1994) Diffusion and sorption in ethylene-propylene copolymers: comparison of experimental methods. Ind Eng Chem Res 33:2483–2491. https://doi.org/10.1021/ie00034a033

    Article  CAS  Google Scholar 

  29. Zielinski JM, Duda JL (1992) Predicting polymer/solvent diffusion coefficients using free-volume theory. AIChE J 38:405–415. https://doi.org/10.1002/aic.690380309

    Article  CAS  Google Scholar 

  30. Minelli M, Sarti GC (2017) Elementary prediction of gas permeability in glassy polymers. J Memb Sci 521:73–83. https://doi.org/10.1016/j.memsci.2016.09.001

    Article  CAS  Google Scholar 

  31. Minelli M, Sarti GC (2017) Thermodynamic modeling of gas transport in glassy polymeric membranes. Membranes (Basel) 7:46. https://doi.org/10.3390/membranes7030046

    Article  CAS  PubMed  Google Scholar 

  32. Vrentas JS, Vrentas CM (1994) Solvent self-diffusion in rubbery polymer-solvent systems. Macromolecules 27:4684–4690. https://doi.org/10.1021/ma00095a007

    Article  CAS  Google Scholar 

  33. Van Krevelen DW, Te Nijenhuis K (2009) Properties of polymers, 4th ed., Elsevier B.V. https://www.elsevier.com/books/properties-of-polymers/van-krevelen/978-0-08-054819-7. Accessed 16 Dec 2022

  34. Peng F, Jiang Z, Hoek EMV (2011) Tuning the molecular structure, separation performance and interfacial properties of poly ( vinyl alcohol )– polysulfone interfacial composite membranes. J Memb Sci 368:26–33. https://doi.org/10.1016/j.memsci.2010.10.056

    Article  CAS  Google Scholar 

  35. Neogi P (1996) Diffusion polymers. In: Neogi P (ed) Marcel Dekker. Marcel Dekker Incorporation, First, pp 143–171

    Google Scholar 

  36. Losi GU, Knauss WG (1992) Free volume theory and nonlinear thermoviscoelasticity. Polym Eng Sci. https://doi.org/10.1002/pen.760320806

    Article  Google Scholar 

  37. Fillers RW, Tschoegl NW (1977) Effect of pressure on the mechanical properties of polymers. J Rheol (N. Y. N. Y) 21:51–100. https://doi.org/10.1122/1.549463

    Article  CAS  Google Scholar 

  38. Vrentas JS, Vrentas CM (2013) Diffusion and mass transfer. CRC Press, New York. https://www.routledge.com/Diffusion-and-Mass-Transfer/Vrentas-Vrentas/p/book/9781466515680. Accessed 16 Dec 2022

  39. Doolittle AK (1951) Studies in newtonian flow. II. the dependence of the viscosity of liquids on free-space. J Appl Phys 22:1471–1475. https://doi.org/10.1063/1.1699894

    Article  CAS  Google Scholar 

  40. Sanchez IC, Lacombe RH (1976) An elementary molecular theory of classical fluids. Pure fluids. J Phys Chem 80:2352–2362. https://doi.org/10.1021/j100562a008

    Article  CAS  Google Scholar 

  41. Nasehzadeh A, Mohseni M, Azizi K (2002) The effect of temperature on the Lennard-Jones (6–12) pair potential function. J Mol Struct THEOCHEM 589–590:329–335. https://doi.org/10.1016/S0166-1280(02)00188-4

    Article  Google Scholar 

  42. Bokis CP, Donohue MD (1995) A closed-form equation of state for Lennard- Jones molecules based on perturbation theory. J Phys Chem 99:12655–12660. https://doi.org/10.1021/j100033a044

    Article  CAS  Google Scholar 

  43. Pionteck J, Pyda M (2014) Landolt-Bornstein: numerical data and functional relationships in science and technology - new series, new series, Springer, Berlin.https://doi.org/10.1007/978-3-642-41542-5

  44. Minelli M, Campagnoli S, De Angelis MG, Doghieri F, Sarti GC (2011) Predictive model for the solubility of fluid mixtures in glassy polymers. Macromolecules 44:4852–4862. https://doi.org/10.1021/ma200602d

    Article  CAS  Google Scholar 

  45. Rodgers PA (1993) Pressure–volume–temperature relationships for polymeric liquids: a review of equations of state and their characteristic parameters for 56 polymers. J Appl Polym Sci 48:1061–1080. https://doi.org/10.1002/app.1993.070480613

    Article  CAS  Google Scholar 

  46. Zielinski JM (1992) Free-volume parameter estimations for polymer / solvent diffusion coefficient predictions. The Pennsylvania State University

  47. Zielinski JM, Duda JL (1992) Influence of concentration on the activation energy for diffusion in polymer-solvent systems. J Polym Sci Part B Polym Phys 30:1081–1088. https://doi.org/10.1002/polb.1992.090301002

    Article  CAS  Google Scholar 

  48. Kamiya Y, Naito Y, Bourbon D (1994) Sorption and partial molar volumes of gases in poly(ethylene-co-vinyl acetate). J Polym Sci Part B Polym Phys 32:281–286. https://doi.org/10.1002/polb.1994.090320209

    Article  Google Scholar 

  49. Kucukpinar E, Doruker P (2006) Molecular simulations of gas transport in nitrile rubber and styrene butadiene rubber. Polymer (Guildf) 47:7835–7845. https://doi.org/10.1016/j.polymer.2006.08.062

    Article  CAS  Google Scholar 

  50. Dobre T, Pârvulescu OC, Sanchez-Marcano J, Stoica A, Stroescu M, Iavorschi G (2011) Characterization of gas permeation through stretched polyisoprene membranes. Sep Purif Technol 82:202–209. https://doi.org/10.1016/j.seppur.2011.09.019

    Article  CAS  Google Scholar 

  51. Whitley DM, Adolf DB (2012) Investigating the permeability of atmospheric gases in polyisobutylene membranes via computer simulation. J Memb Sci 415–416:260–264. https://doi.org/10.1016/j.memsci.2012.05.008

    Article  CAS  Google Scholar 

  52. Lundberg JL, Mooney EJ (1969) Diffusion and solubility of methane in polyisobutylene. J Polym Sci Part A 7:947–962. https://doi.org/10.1002/pol.1969.160070517

    Article  CAS  Google Scholar 

  53. Garzón B, Lago S, Vega C, De Miguel E, Rull LF, Garzon B, Lago S, Vega C, De Miguel E, Rull LF (2013) Computer simulation of vapor – liquid equilibria of linear quadrupolar fluids. Departures from the principle of corresponding states Computer simulation of vapor-liquid equilibria of linear quadrupolar fluids. Departures from the principle of correspond. J Chem Phys. https://doi.org/10.1063/1.467467

    Article  PubMed  Google Scholar 

  54. Naito Y, Kamiya K (1989) Sorption and partial molar volumes of gases in polybutadiene. Metab Eng 4:2243–2250. https://doi.org/10.1002/polb.1989.090271107

    Article  Google Scholar 

  55. Stark W, Jaunich M (2011) Investigation of ethylene/vinyl acetate copolymer (EVA) by thermal analysis DSC and DMA. Polym Test 30:236–242. https://doi.org/10.1016/j.polymertesting.2010.12.003

    Article  CAS  Google Scholar 

  56. Skoulidas AI, Sholl DS (2005) Self-diffusion and transport diffusion of light gases in metal-organic framework materials assessed using molecular dynamics simulations. J Phys Chem B 109:15760–15768. https://doi.org/10.1021/jp051771y

    Article  CAS  PubMed  Google Scholar 

  57. Buckingham AD, Longuet-Higgins HC (1968) The quadrupole moments of dipolar molecules. Mol Phys 14:63–72. https://doi.org/10.1080/00268976800100051

    Article  CAS  Google Scholar 

  58. Zarshenas K, Raisi A, Aroujalian A (2016) Mixed matrix membrane of nano-zeolite NaX/poly (ether-block-amide) for gas separation applications. J Memb Sci 510:270–283. https://doi.org/10.1016/j.memsci.2016.02.059

    Article  CAS  Google Scholar 

  59. Matteucci S, Raharjo RD, Kusuma VA, Swinnea S, Freeman BD (2008) Gas permeability, solubility, and diffusion coefficients in 1,2-polybutadiene containing magnesium oxide. Macromolecules 41:2144–2156. https://doi.org/10.1021/ma702459k

    Article  CAS  Google Scholar 

  60. Mulder M (1996) Basic principles of membrane technology. Second, Kluwer Academic Publishers. https://doi.org/10.1007/978-94-009-1766-8

  61. Tan JH, Chen CL, Liu YW, Wu JY, Wu D, Zhang X, She ZH, He R, Zhang HL (2020) Molecular simulations of gas transport in hydrogenated nitrile butadiene rubber. J Polym Res. https://doi.org/10.1007/s10965-020-02258-3

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Hafez Ave., P.O. Box 15875-4413, Tehran, Iran

    Mohammad Sajad Sepehri Sadeghian & Ahmadreza Raisi

Authors
  1. Mohammad Sajad Sepehri Sadeghian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ahmadreza Raisi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Mohammad Sajad Sepehri Sadeghian: Conceptualization, Methodology, Data collection, Writing—original draft preparation. Ahmadreza Raisi: Supervision, Study design, Data analysis and interpretation, Writing—review and editing.

Corresponding author

Correspondence to Ahmadreza Raisi.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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

Sepehri Sadeghian, M.S., Raisi, A. A predictive procedure to model gas transport and intrinsic properties of rubbery polymeric membranes using equilibrium thermodynamics and free volume theory. J Polym Res 30, 98 (2023). https://doi.org/10.1007/s10965-023-03482-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10965-023-03482-3

Keywords

Search

Navigation