# Theoretical and Experimental Considerations for Investigating Multicomponent Diffusion in Hydrated, Dense Polymer Membranes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}capture, where O

_{2}, N

_{2}, CO

_{2}and other species may be present. In water desalination applications, multiple different salts (NaCl, CaCl, MgCl) are present, while in electrochemical applications, different mixtures of electrolyte as well as feed and product molecules/ions may be present in the device. In this review, we narrow our focus to hydrated, dense polymer membranes and highlight considerations for experimentally investigating solution diffusion transport for multicomponent systems.

_{i}, that is defined as the product of the kinetic diffusivity, D

_{i}, and the thermodynamic solubility, K

_{i}:

_{i}, can be measured experimentally using a diffusion cell apparatus (Figure 1a). A typical diffusion cell apparatus consists of a donor cell and receiver cell sandwiched around a hydrated membrane of known thickness. The receiver cell solution typically begins as pure deionized water (DI water) while the donor cell solution contains the solute(s) of interest in known concentration. Over time, the solute(s) diffuse across the membrane from the donor cell to the receiver cell down the concentration gradient(s) of the solute(s), shown in Figure 1b.

^{®}that was similar to that reported elsewhere using other techniques, while also uncovering previously unreported differences between single-solute and multicomponent transport for mixtures of methanol and two carboxylates (sodium acetate and sodium formate). Dobyns et al. [12] used in situ ATR FTIR spectroscopy to study diffusion of multicomponent solutions of methanol and sodium acetate in membranes made from poly(ethylene glycol) diacrylate (PEGDA). In that work, the permeability to methanol increased in the presence of sodium acetate as a co-solute while the permeability to sodium acetate decreased, compared to single-solute measurements. Subsequently, Kim et al. [24] incorporated 2-acrylamido-2-methylpropanesulfonic acid (AMPS) as a comonomer with PEGDA to fabricate crosslinked cation exchange membranes and again found differences between single-solute and multicomponent permeabilities of methanol and sodium acetate. These effects are not restricted to alcohol/carboxylate solutions; for example, differences between single-solute and multicomponent permeabilities have also been observed for solutions of different alcohols. Dobyns et al. [25] investigated the permeabilities of solutions containing up to four solutes (methanol, ethanol, n-propanol, and acetone) for the commercial cation exchange membrane Nafion

^{®}117. The membrane selectivity to an aqueous solution of ethanol and n-propanol varied by up to 60% compared to single-solute measurements.

## 2. Modeling of Multicomponent Diffusion

#### 2.1. Solution-Diffusion Model

_{s}solutes is treated as a single thermodynamic phase. The pressure within this phase is constant and equal to that of the feed solution; however, the permeate solution can be at a lower pressure, e.g., in reverse osmosis processes. This pressure drop—modeled as a step change at the permeate boundary (Figure 1b)—must be mechanically supported, for example, by a porous layer. Because the pressure is constant within the membrane, transport of solutes is driven by diffusion due to concentration gradients. Typically, the solute composition in the feed is prescribed, and the composition of the permeate solution is determined by a combination of thermodynamic and transport considerations that determine the membrane’s performance.

_{i}

_{,f}is the concentration of component i in the feed solution (Figure 1b). Note that K

_{i}is a thermodynamic property that generally depends on composition, both in the membrane and in the feed solution, but is frequently assumed to be a constant for a given solute–membrane pair. However, in multicomponent solutions, K

_{i}may increase or decrease due to the presence of other solutes that enhance or reduce solubility of component i, respectively, in ways that are challenging to anticipate.

_{i}is the average velocity of component i. The diffusive flux ${j}_{i}={c}_{i}\left({v}_{i}-v\right)$ is defined as the flux relative to advection at a reference velocity v. Different reference velocities can be useful for analysis in different situations. For example, a common choice of v for bulk solutions is the barycentric (mass-averaged) velocity, $v={{\displaystyle \sum}}_{i}{\omega}_{i}{v}_{i}$ where ${\omega}_{i}$ is the mass fraction of component i, but the mole-averaged velocity (using the mole fraction x

_{i}of component i instead of ${\omega}_{i}$) and the volume-averaged velocity (using the volume fraction ${\varphi}_{i}$ of component i instead of ${\omega}_{i}$) are also used. For membranes, it can be particularly helpful to define $v={v}_{\mathrm{m}}$, where v

_{m}is the velocity of the membrane itself, because the membrane is stationary in the laboratory frame (${v}_{\mathrm{m}}=0$) while the barycentric velocity may be nonzero and unknown.

_{i}, can be modeled using several approaches, each parametrized by different types of diffusion coefficients. Care must be taken when applying these models because some diffusion coefficients depend on the reference velocity. A simple model for j

_{i}is Fick’s law, which when defined relative to the barycentric velocity is [30]

_{i}is the molecular weight of component i, and D

_{i}is the mutual (Fick) diffusion coefficient of component i. If $\rho $ is independent of composition, the familiar expression ${j}_{i}=-{D}_{i}\partial {c}_{i}/\partial x$ is obtained.

_{i}. Assuming both $\rho $ and D

_{i}are independent of composition, the steady-state flux across the membrane is

_{i}

_{,p}is the concentration of component i in the permeate solution, and we have assumed the same solubility, K

_{i}, on the feed and permeate boundaries. The permeability, P

_{i}, of component i can be obtained from experimental measurements of c

_{i}

_{,p}in the receiver cell. For example, the commonly used Yasuda model, which can be obtained from Equation (3) [31], is

_{p}is the volume of solution in the receiver cell. The Fick diffusion coefficient, D

_{i}, can then be extracted from P

_{i}if the solubility, K

_{i}, is known.

_{i}and D

_{i}extracted in this way can show complex dependencies on composition that are challenging to anticipate or interpret (Section 1). These dependencies need not be the same for both P

_{i}and D

_{i}, as P

_{i}includes the composition dependence of K

_{i}but D

_{i}does not. Moreover, D

_{i}is implicitly a pseudobinary diffusion coefficient associated with another dominant component such as the membrane and using this model for a multicomponent solution effectively neglects coupled transport due to solute–solute interactions.

#### 2.2. Multicomponent Diffusion

_{ik}is the Onsager coefficient coupling the flux of component i to the gradient of the chemical potential, ${\mu}_{k}$, of component k and T is the temperature. Not all L

_{ik}are independent: the Onsager coefficients can be written as a matrix that must be symmetric and positive semi-definite as consequences of microscopic reversibility and the second law of thermodynamics. Further, the diffusive fluxes are defined relative to the reference velocity v so only $n-1$ of the fluxes j

_{i}are independent, e.g., for the barycentric reference velocity, ${{\displaystyle \sum}}_{i=1}^{n}{M}_{i}{j}_{i}=0$. Simultaneously, the chemical potential gradients are constrained by the Gibbs–Duhem relationship (assuming local thermodynamic equilibrium),

_{ik}does not have typical dimensions of a diffusion coefficient; an Onsager diffusion coefficient, ${\mathsf{\Lambda}}_{ik}={L}_{ik}{k}_{\mathrm{B}}/c$ where $c={{\displaystyle \sum}}_{i=1}^{n}{c}_{i}$ is the total concentration, is sometimes defined [36]. The Onsager coefficients are also phenomenological and depend on the reference velocity so, as a result, can be challenging to interpret.

_{B}is the Boltzmann constant. The Maxwell–Stefan diffusion coefficients, Đ

_{ik}, are independent of the reference velocity, and, with some algebra, can be related directly to the Onsager coefficients [36,37,38]. Both Equations (5) and (7) use the chemical potential gradient as a driving force for diffusion, but these are not directly measurable in experiments and must be estimated from composition gradients using an activity-coefficient model. This makes the Onsager and Maxwell–Stefan diffusion coefficients challenging to determine.

_{ik}, depend on the reference velocity. Equation (8) can be shown to systematically follow from Equations (5) and (7). A consequence of this is that the multicomponent Fick diffusion coefficients, D

_{ik}, can be decomposed into a dynamic contribution, related to the Onsager or Maxwell–Stefan diffusion coefficients, and a thermodynamic contribution, based on an activity-coefficient model for the chemical potentials.

#### 2.3. Simulating Multicomponent Diffusion Coefficients

_{ik}can then be determined by analyzing the trajectory using [36,37,38,39,40]

_{ik}are determined, the Maxwell–Stefan diffusion coefficients can be directly computed. The multicomponent Fick diffusion coefficients require an additional thermodynamic model.

## 3. Experimental Approaches to Investigating Multicomponent Diffusion

_{i}, is modeled as a simple product of the solubility, K

_{i}, and diffusivity, D

_{i}(Equation (1)). Thus, a common experimental approach for determining the diffusivity of the solutes is to (1) measure the permeabilities of the solutes through the membrane (e.g., via a diffusion cell experiment), (2) measure the solubility of the solutes through the membrane via an independent sorption–desorption experiment, and then (3) calculate the diffusivities. Here, we examine the experiments used to determine the permeability and solubility of a solute in a hydrated, dense polymer membrane.

^{®}117 by circulating the receiver cell solution through a benchtop ATR FTIR spectrometer. A later study by Carter et al. [59] on multicomponent transport of alcohols through Selemion AMV removed the need for recirculation, along with the accompanying time-delay and leakage issues.

_{0}are the transmitted and incident intensity of light, and l is the incident light’s path length which travels through the solution. This relationship can be written compactly by defining the effective molar absorptivity ${\u03f5}_{\lambda}={E}_{\lambda}l$. If the solution contains multiple species, the Beer–Lambert law is additive,

_{s}is the number of solutes. Both the solutes and solvent are included in this summation (n

_{s}+1) as the general case since the concentration of solvent changes corresponding to the concentration of solutes. However, this is usually accounted for through background subtraction of the receiver cell absorbance at t = 0 and by choosing wavenumbers for the solutes that are unaffected by the solvent spectra.

^{−1}for methanol, 1414 cm

^{−1}for sodium acetate, and 1350 cm

^{−1}for sodium formate) [23]. Figure 3 shows an example of the calibration process for methanol and sodium acetate. Once the molar absorptivity for each of the solute at a certain wavenumber are obtained, they are utilized to determine the concentrations from absorbance spectra. Time-resolved concentration data then yields the permeability of the membrane to each solute.

_{i}, by performing a sorption–desorption experiment, shown schematically in Figure 4. Briefly, these experiments involve equilibrating the membrane with a donor solution and subsequently desorbing in order to determine how much of the solute(s) were taken up by the membrane by measuring the concentration of the solute(s) in the desorption solution using an appropriate method (HPLC, conductivity, etc.) The solubility, K

_{i}, is then calculated as the ratio of the concentration of solute in the membrane and the concentration of the donor solution. This allows for the diffusivity, D

_{i}, to then be calculated from the solution-diffusion model (Equation (1)).

## 4. Future Outlook and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

P_{i} | Permeability |

D_{i} | Fick’s law diffusivity |

K_{i} | Solubility |

n | Number of components |

n_{s} | Number of solutes |

c_{i} | Concentration of component i inside the membrane |

c_{i,f} | Concentration of component i in feed solution |

c_{i,p} | Concentration of component i in permeate solution |

v_{i} | Average velocity of component i |

v | Reference velocity |

j_{i} | Diffusive flux relative to reference velocity |

$\omega $_{i} | Mass fraction of component i |

M_{i} | Molecular weight of component i |

$\rho $ | Total mass density |

${\mu}_{i}$ | Chemical gradient of component i |

L_{ij} | Onsager coefficient |

${\Lambda}_{ij}$ | Onsager diffusion coefficient |

Đ_{ik} | Maxwell Stefan diffusion coefficient |

D_{ij} | Multicomponent Fick diffusion coefficient |

${A}_{\lambda}$ | Absorbance at wavenumber λ |

${E}_{\lambda}$ | Molar absorptivity of the solute at wavenumber λ |

I | Transmitted light intensity |

I_{o} | Incident light intensity |

${\u03f5}_{\lambda}$ | Effective molar absorptivity |

l | Light’s path length |

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**Figure 1.**(

**a**) Depiction of the diffusion cell apparatus using an in situ ATR FTIR spectroscopy probe (Section 3) and (

**b**) concentration and pressure gradients throughout the membrane according to the solution-diffusion model.

**Figure 2.**Schematic of multicomponent permeability measurement through a polymeric membrane using a diffusion cell coupled with in situ ATR-FTIR spectroscopy.

**Figure 3.**Absorbance data obtained from in situ ATR FTIR plotted against various concentrations in ultrapure water for (

**a**) methanol, (

**b**) sodium acetate, at various wavenumbers showing desired linear relationship. Reprinted (adapted) with permission [23]. Copyright 2022 Elsevier.

**Figure 4.**Schematic diagram of sorption-desorption experiment. (Left to right) Equilibration in DI water, Equilibration with solution of solute (s) of interest, followed by successive equilibrium desorption in DI water with solution concentrations determined by HPLC.

Model | Driving Force | Transport Coefficient | Notes |
---|---|---|---|

Solution-diffusion model with Fick’s law (Equation (3)) | Composition gradient | P_{i} | Based on Fick’s law for diffusion (Equation (2)), which neglects off-diagonal (i ≠ j) fluxes in Equation (8). P_{i} is the product of the diffusivity, D_{i}, and the solubility, K_{i}. |

Multicomponent Fick’s law (Equation (8)) | Composition gradient | D_{ij} | D_{ij} can be related to L_{ij} or Đ_{ij} using a thermodynamic model. |

Nonequilibrium thermodynamics (Equation (5)) | Chemical potential gradient | L_{ij} or ${\mathsf{\Lambda}}_{ij}$ | L_{ij} are measurable in equilibrium molecular dynamics simulations (Equation (9)). |

Maxwell–Stefan (Equation (7)) | Chemical potential gradient | Đ_{ij} | Đ_{ij} are independent of reference velocity, can be computed from L_{ij}. |

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 (D_{i}). 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 (P_{i}). 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 (P_{i}). |

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 (K_{i}) to the solute(s). |

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Mazumder, 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