Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes
Abstract
:1. Introduction
2. Modeling of Multicomponent Diffusion
2.1. Solution-Diffusion Model
2.2. Multicomponent Diffusion
2.3. Simulating Multicomponent Diffusion Coefficients
3. Experimental Approaches to Investigating Multicomponent Diffusion
4. Future Outlook and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
Pi | Permeability |
Di | Fick’s law diffusivity |
Ki | Solubility |
n | Number of components |
ns | Number of solutes |
ci | Concentration of component i inside the membrane |
ci,f | Concentration of component i in feed solution |
ci,p | Concentration of component i in permeate solution |
vi | Average velocity of component i |
v | Reference velocity |
ji | Diffusive flux relative to reference velocity |
i | Mass fraction of component i |
Mi | Molecular weight of component i |
Total mass density | |
Chemical gradient of component i | |
Lij | Onsager coefficient |
Onsager diffusion coefficient | |
Đik | Maxwell Stefan diffusion coefficient |
Dij | Multicomponent Fick diffusion coefficient |
Absorbance at wavenumber λ | |
Molar absorptivity of the solute at wavenumber λ | |
I | Transmitted light intensity |
Io | Incident light intensity |
Effective molar absorptivity | |
l | Light’s path length |
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Model | Driving Force | Transport Coefficient | Notes |
---|---|---|---|
Solution-diffusion model with Fick’s law (Equation (3)) | Composition gradient | Pi | Based on Fick’s law for diffusion (Equation (2)), which neglects off-diagonal (i ≠ j) fluxes in Equation (8). Pi is the product of the diffusivity, Di, and the solubility, Ki. |
Multicomponent Fick’s law (Equation (8)) | Composition gradient | Dij | Dij can be related to Lij or Đij using a thermodynamic model. |
Nonequilibrium thermodynamics (Equation (5)) | Chemical potential gradient | Lij or | Lij are measurable in equilibrium molecular dynamics simulations (Equation (9)). |
Maxwell–Stefan (Equation (7)) | Chemical potential gradient | Đij | Đij are independent of reference velocity, can be computed from Lij. |
Experimental Approach | Variables to Measure | Significance |
---|---|---|
Interferometry | Measures refractive index variation in liquid layers adjacent to membrane | Measured refractive index used to calculate diffusivity coefficients (Di). Experiments and calculations are quite complex. |
Diffusion-Cell experiments with aliquotic sampling | Measures solute(s) concentration(s) in receiver cell utilizing ex situ spectroscopic methods | Measured solute(s) concentration(s) used in Yasuda model to determine permeability (Pi). For multicomponent systems sampling results in non- constant volume in cell and is aliquot analysis time-consuming. |
Diffusion-Cell experiments coupled with in situ ATR FTIR | Measures solute (s) concentration(s) variation in receiver cell using in situ ATR FTIR spectroscopy. | Real time concentration data obtained from the diffusion cell for use in Yasuda’s model to extract multi-solute permeabilities (Pi). |
Sorption-desorption experiment | Measures concentration of solute desorbed from membrane after equilibrium sorption. | Desorbed solute(s) concentration(s) used with measured membrane volume to calculate membrane solubility (Ki) to the solute(s). |
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Mazumder, A.; Dobyns, B.M.; Howard, M.P.; Beckingham, B.S. Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes. Membranes 2022, 12, 942. https://doi.org/10.3390/membranes12100942
Mazumder A, Dobyns BM, Howard MP, Beckingham BS. Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes. Membranes. 2022; 12(10):942. https://doi.org/10.3390/membranes12100942
Chicago/Turabian StyleMazumder, Antara, Breanna M. Dobyns, Michael P. Howard, and Bryan S. Beckingham. 2022. "Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes" Membranes 12, no. 10: 942. https://doi.org/10.3390/membranes12100942
APA StyleMazumder, A., Dobyns, B. M., Howard, M. P., & Beckingham, B. S. (2022). Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes. Membranes, 12(10), 942. https://doi.org/10.3390/membranes12100942