# Models for Facilitated Transport Membranes: A Review

^{*}

## Abstract

**:**

## 1. Introduction

_{3}salt as liquid membranes, supported on tetrafluoroethylene (PTFE), by Azizi et al. [17] to separate propylene from propane in 2015. Supported ionic liquid membranes have been recently used by Zarca and co-workers [18] to enhance the selectivity of propylene/propane. FSC systems based on polymer incorporating silver nanoparticles are under study in olefin separation, to overcome the chemical stability issues of the silver ion [19,20,21,22,23].

_{2}before the scrubber unit.

_{2}separation from other gases, such as N

_{2}(post-combustion), CH

_{4}(biomethane and natural gas upgrading), H

_{2}(steam reforming purification) [31,32,33]. The massive anthropogenic emissions of the last decades are changing dramatically the environment in terms of global temperature raise, oceans acidification, as well as Arctic and Antarctic defrost and sea level rise [34]. Facilitated transport membranes could be an interesting, promising, ecofriendly alternative to the common solutions, such as amino based-separations unity. Guha et al. [35] in 1990 used an aqueous solution of diethanol amine (DEA) immobilized in poly-propylene microporous support, Davis and Sandall also used DEA aqueous solution but immobilized in poly-ethylene glycol, in 1993 [36]. Teramoto, in 1996, used poly-vinyl diene difluoride as supporting polymer for DEA and monoethanol amine (MEA) solutions [37]. FSC membranes have been extensively tested in the CO

_{2}separation in recent years. Matsuyama et al. used poly-ethylene imine blended with poly-vinyl alcohol [38], and Ho and co-workers studied mobile-fixed hybrid carrier systems made by poly-allyl amine/poly-vinyl alcohol blend with amino acid and potassium hydroxide in 2006 and 2008, [39,40]. Deng et al. [41] presented their results about the facilitated transport of carbon dioxide in poly-vinyl amine and poly-vinyl alcohol blend membranes in 2009. Poly-vinyl amine membranes were also tested in a pilot scale plant in 2013 by the same research group [42] showing stability of performance over four months of test and another pilot scale test was conducted in 2017 and reported here, [43]. Nowadays, concerning FTM for CO

_{2}separations, the hybrid systems containing nanofillers [44,45,46] as well as mobile and fixed sites carrier, such as ionic liquids or amino acid salts or polymers containing suitable pendant groups, [47,48,49,50,51,52] are investigated to increase membrane stability and at the same time to obtain high selectivity and permeability.

_{2}facilitated transport in a poly-vinyl alchol (PVA)-amines membrane by the finite element method. Because of the lack of exact analytical solutions, the reaction-augmented transport problem still represents an interesting topic in mathematical and computational studies in the differential nonlinear systems field [59,62,63,64,65,66,67].

#### Mathematical Background

- C = concentration
- J = flux
- r = dissipative term
- r’ = generative term
- v = velocity
- x, y, z = cartesian directions
- t = time

_{AC}. Both kinetic constants, ${k}_{f}$ and ${k}_{r}$, are considered concentration-independent.

_{1}and C

_{2}are integration constants, and C

_{1}in particular represents the total solute flux.

## 2. Mobile Carrier Systems

#### 2.1. Models for Mobile Carrier Facilitated Transport Membranes

#### 2.1.1. Friedlander and Keller, 1965. Mass Transfer in Reacting System Near Equilibrium: Use of the Affinity Function

#### 2.1.2. Blumenthal and Katchalsky, 1969. The Effect of Carrier Association–Dissociation Rate on Membrane Permeation

#### 2.1.3. Goddard et al., 1969. On Membrane Diffusion with Near Chemical Equilibrium Reaction

_{eq}truncated to the first term of the expansion:

_{eq}is the one derived in the mathematical background and hereafter again shown:

#### 2.1.4. Kreuzer and Hoofd, 1970. Facilitated Diffusion of Oxygen in the Presence of Hemoglobin

_{1}, E

_{1}, and B are reported in the supporting materials.

#### 2.1.5. Kreuzer and Hoofd, 1972, Factors Influencing Facilitated Diffusion of Oxygen in the Presence of Hemoglobin and Myoglobin

#### 2.1.6. Yung and Probstein, 1973. Similarity Considerations in Facilitated Transport

#### 2.1.7. Smith et al., 1973. An Analysis of Carrier Facilitated Transport

#### 2.1.8. Smith, Quinn, 1979. The Prediction of Facilitation Factors for Reaction-Augmented Membrane Transport

_{2}in enzymatically bounded polymer in particular conditions. That work has not been discussed in this review, because the reaction scheme used deviates from the one taken as case of study here.

_{eq}the facilitation effect disappears and F tends to unity due to a saturation of the carrier agent (Schultz et al. [1]). This phenomenon implies that there is a value of the equilibrium constant that maximizes the facilitation, as explained in detail by Kemena et al. [7]. On the other hand, in the solution of Smith and Quinn presented here, the facilitation factor reaches a plateau by increasing the equilibrium constant value, as shown in Figure 5.

#### 2.1.9. Noble et al., 1986. Effect of Mass Transfer Resistance on Facilitated Transport

#### 2.1.10. Basaran et al., 1989. Facilitated Transport with Unequal Carrier and Complex Diffusivities

#### 2.1.11. Jemaa and Noble, 1992. Improved Analytical Prediction of Facilitation Factors in Facilitated Transport

#### 2.1.12. Teramoto, 1994. Approximate Solution of Facilitation Factor in Facilitated Transport

#### 2.1.13. Morales-Cabrera et al., 2002. Approximate Method for the Solution of Facilitated Transport Problems in Liquid Membranes

## 3. Fixed Carrier Systems

#### 3.1. Models for Fixed Sites Facilitated Transport Membranes

_{2}in polymeric membranes having aminoethoxycarbonyl moieties (Yoshikawa et al. [106,107]). Moreover, the dual mode comprehensive transport framework derived by Barrer [108] in 1984, was used by Noble [109,110,111] to derive some crucial results about the fixed carrier facilitated transport. Even if the dual mode has been applied to many fixed carrier systems, with good results in terms of fitting, this theory does not provide a clear and full explanation of the transport mechanism in these facilitated membranes, in particular regarding the chemical reaction effect.

#### 3.1.1. Dual Mode Theory

- ${C}_{A}$ = solute concentration in polymer phase
- ${C}_{H}$ = solute concentration trapped in ‘holes’
- ${C}_{D}$ = solute concentration dissolved
- $p$ = solute partial pressure
- ${k}_{d}$ = Henry’s constant
- $C{H}^{\prime}$ = holes saturation level concentration
- $b$ = affinity constant

_{2}and N

_{2}, the permeability was independent from the upstream pressure, for the carbon dioxide the behavior was quite different. Their data, shown below in Figure 9a, evidence a decrease in permeability as the upstream pressure increases, in agreement with the saturation mechanism explained. Even in this case, the experimental permeability of carbon dioxide was modeled by the partial immobilization model of Paul and Koros, Equation (164) [120].

#### 3.1.2. Cussler et al., 1989. On the Limits of Facilitated Diffusion

_{0}is the amplitude of the oscillation. If the distance between equilibrium locations is higher than the oscillation width, the carriers cannot get in touch and no solute exchange is possible. In this case, no diffusion flux can be observed. Conversely, if ${l}_{0}\ge l$ two carriers could get in contact and an exchange mechanism can take place. The latter condition is therefore necessary for the solute transport. If that condition is achieved, the equation for the flux is:

#### 3.1.3. Noble, 1990. Analysis of Facilitated Transport with Fixed Site Carrier Membranes

_{C}, is nearly constant and not too different from the total carrier concentration C

_{T}, Noble found that Equation (13) describes the complex mass balance also in the fixed carrier case. The derivation of such equation requires the existence of the four exchange mechanisms introduced by Barrer, [108], conversely to what Cussler did [100].

_{AC}, is now function of the morphology of the matrix and of the mobility between sites. That function was found to be in the form:

- -
- the mathematical derivation of the mass balance analog of the mobile carrier case, which allows to use, in analytical approximation methods already known, Equation (2.89) while an excess of carrier is considered.
- -
- the functional dependence of the actual complex diffusivity in such systems on morphological and chemical parameters, Equation (182).

#### 3.1.4. Noble, 1991, Facilitated Transport Mechanism in Fixed Site Carrier Membranes

_{hh}in Barrer notation Equation (168), is equal to zero.

#### 3.1.5. Noble, 1992, Generalized Microscopic Mechanism of Facilitated Transport in Fixed Site Carrier Membranes

#### 3.1.6. Kang et al., 1995. Analysis of Facilitated Transport in Solid Membranes with Fixed Site Carriers

#### 3.1.7. Zarca et al., 2017. A Practical Approach to Fixed-Site Carrier Facilitated Transport Modeling for the Separation of Propylene/Propane Mixtures Through Silver-Containing Polymeric Membranes

_{4}embedded in polymeric membranes (PVDF-HFP) with an analytical approach. Furthermore, that approach was extended to consider also the presence of mobile carriers inside the membranes [18] (see next paragraph).

_{dd}in Noble’s notation, and the second one incorporates all the effects of the chemical reaction. The exchange between different zones of the membrane has been not considered explicitly, and a hopping mechanism was hypothesized.

#### 3.1.8. Zarca et al., 2017. Generalized Predictive Modeling for Facilitated Transport Membranes Accounting For Fixed and Mobile Carriers

_{4}, was also incorporated in the poly-vinyldene fluoride-co-hexafluoropropylene (PVDF-HFP) polymeric membranes, together with a silver salt, AgBF

_{4}. The resulting transport mechanism is a combination of the two effects of mobile and fixed carrier. Silver ions bounded to the polymer backbone act as fixed carriers while, the unbounded ones, due to the presence of liquid ion, are free to diffuse in the system, acting as mobile carrier. For these hybrid systems the total solute flux has the form:

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

^{2}“NanoMaterials Enhanced Membranes for Carbon Capture”, funded by the Innovation and Networks Executive Agency (INEA) Grant Agreement Number: 727734.

## Conflicts of Interest

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**Figure 1.**Effect of ratio between membrane thickness (L) and characteristic length (λ) on the transport in the model of Blumenthal and Katchalsky. (

**a**) Size of equilibrium core in the membranes for different λ values (Continuous lines λ = 0.05, dotted lines λ = 0.1, broken lines λ = 0.2, broken dotted lines λ = 1); (

**b**) Facilitation factor as function of λ. F

_{eq}represents Equation (40).

**Figure 2.**Model of Kreuzer and Hoofd versus experimental data by Wittemberg [5]. (

**a**) Oxygen facilitated flux in function of hemoglobin concentration; (

**b**) oxygen facilitated flux in function of the oxygen pressure at the feed side. In both figures the points are the experimental data, the line reports the model prediction. The flux reported is the reactive component of the total flux.

**Figure 3.**Comparison between facilitation factor F calculated numerically by Kutchai et al. [53] and the value calculated by Equation (86) proposed by Yung and Probstein [90] for different values of $\u03f5$ and $\kappa $ and Δ fixed. (

**a**) $\kappa $ = 4.5, Δ = 4.5; (

**b**) $\kappa $ = 35.9, Δ = 1122. Dashed lines represent the parity condition.

**Figure 5.**Model prediction of facilitated factor F. (

**a**) Numerical results of Kutchai et al. [53] and Equation (101) calculations; (

**b**) Asymptotic behavior of Equation (101) as function of dimensionless equilibrium constant (K = K

_{eq}C

_{A}

^{0}). Symbols are numerical results from Kemena et al. [7], filled line is Equation (101), dotted line is the improved model of Jeema and Noble, Equation (126).

**Figure 6.**Diffusivity ratio influence on facilitation. (

**a**) Small Damköhler number. Black squares and white circles are numerical results for P = 1 and 0.5 respectively, full and broken lines are Equation (120) for P = 1 and 0.5 respectively; (

**b**) Large Damköhler number. Black squares and white circles are numerical results for P = 1000 and 10,000 respectively. Full line is Equation (124). In both the case small but nonzero downstream concentration has been considered ($\overline{{C}_{A}^{L}}$ = 0.01).

**Figure 7.**Solute concentration profiles. (

**a**) $\delta $ = 22.5, circles are numerical solution from Jain and Shultz [58], solid line is the upstream solution, broken line is the downstream solution, dash dotted line is the equilibrium core solution; (

**b**) $\delta $ = 5, solid line is the upstream solution, broken line is the downstream solution. In both cases the downstream solute concentration is zero.

**Figure 8.**Facilitation factor as function of Damköhler number (Equation (157)). (

**a**) Influence of the downstream solute concentration (dimensionless) for the case D

_{AC}= D

_{C}. From the top, $\overline{{C}_{A}^{1}}$ = 0, 0.1, 0.4. Circles are numerical solution from Kutchai et al. [53], lines are calculations from the present model; (

**b**) diffusivities ratio effect, r = D

_{AC}/D

_{C}

_{,}for zero downstream concentration. From the top, r = 1, r = 0.5, and r = 0.25. Continuous lines are the model of Morales and Cabrera [98], dash and dotted lines are the asymptotic solution of Teramoto [74].

**Figure 9.**Dual mode model results, permeability as function of upstream pressure. (

**a**) Yoshikawa et al. [106]. Circles, crosses and triangles are CO

_{2}, O

_{2}and N

_{2}measured, respectively; (

**b**) Nishide et al. [105]. Lines are the model predictions. Permeability is in (1 × 10

^{−9}× cm

^{3}·cm/(cm

^{2}·s·atm)), pressure is in (atm).

**Figure 10.**Representation of membrane thickness in Cussler et al. [100]. (

**a**) Series arrangement of contiguous lamellae. L is the membrane thickness. (

**b**) Contiguous fixed sites. l

_{0}is the oscillation width, l is the distance between equilibrium positions. For l > l

_{0}solute exchange between contiguous sites cannot occur.

**Figure 11.**Solute flux as function of carrier concentration. (

**a**) Qualitative behavior in presence of a threshold limit. For carrier concentration below the threshold, the solute flux does not occur; (

**b**) experimentally detected fructose flux in plasticized cellulose triacetate as function of carrier (TOMAC) concentration, Riggs and Smith [125].

**Figure 12.**Solute transport pathways in fixed sites carrier systems used by Noble [109].

**Figure 13.**Modeling results using Equation (3.22). (

**a**) Plot of E = (F − 1)

^{−1}as function of upstream pressure. At high pressure, the linearity is lost. (

**b**) Facilitation factor calculated by using parameters retrieved in the low pressure range of figure a. Circles are experimental data from Tsuchida et al. [127], lines come from Equation (183).

**Figure 14.**Membrane environment in the present work of Noble. Between sites, in the length space l

_{2}, pure diffusion mechanism exists. Closely to the sites position, l

_{1}, Equation (184) holds.

**Figure 15.**Analogies between RC parallel circuit and fixed sites carrier systems. (

**a**) Parallel configuration of single RC circuit; (

**b**) fixed sites carrier systems. Analogies among driving forces and capacitance effects according to Table 1.

**Figure 16.**Oxygen permeability in a poly-dimethyl siloxane (PDMS) membrane with metallo-porphyrin fixed carrier. Circles are experimental data from Ohynagi et al. [26], n = 228 and n = 1 for continuous and broken line, respectively. Permeability is in (1 × 10

^{−9}× cm

^{3}·cm/(cm

^{2}·s·atm)), pressure is in (atm).

**Figure 17.**Ethylene flux as function of upstream pressure in membranes of poly-vinyldene fluoride (PVDF) containing AgBF

_{4}as fixed carrier at two different temperatures. Symbols are experimental data from Zarca et al., lines are model results. From the top, T = 303.15 and 293.15 K. Flux is expressed in (1 × 10

^{−6}mol/cm

^{2}·s), pressure is in (atm).

**Figure 18.**Ethylene flux in polymeric membranes (PVDF-HFP)/BMImBF

_{4}/AgBF

_{4}. (

**a**) Influence of ethylene pressure at two different temperature. Symbols are experimental data, lines are model calculations from Zarca et al. [18]. From the top, T = 303.15 K, 293.15 K; (

**b**) Prediction of fixed site (black) and mobile carrier (white) contribution to total flux as function of temperature. Flux is expressed in (1 × 10

^{−6}mol/cm

^{2}·s), pressure is in (atm), temperature is in (K).

**Table 1.**Analogies between electrons transport and mass transport in resistor–capacitor circuits (RC) parallel circuit and facilitated transport systems with fixed sites carrier [112].

RC circuit | Facilitated Systems | |
---|---|---|

Flux | ${j}_{e}=\sigma \frac{\Delta V}{L}$ | $J=P\frac{{p}^{0}}{L}$ |

Driving Force | $\frac{\Delta V}{L}$ | $\frac{{p}^{0}}{L}$ |

Proportionality | $\sigma $ | $P$ |

Capacitor Effect | $q=CV$ | ${C}_{AC}=\frac{{C}_{T}{C}_{A}{K}_{eq}}{1+{C}_{A}{K}_{eq}}$ |

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Rea, R.; De Angelis, M.G.; Baschetti, M.G.
Models for Facilitated Transport Membranes: A Review. *Membranes* **2019**, *9*, 26.
https://doi.org/10.3390/membranes9020026

**AMA Style**

Rea R, De Angelis MG, Baschetti MG.
Models for Facilitated Transport Membranes: A Review. *Membranes*. 2019; 9(2):26.
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**Chicago/Turabian Style**

Rea, Riccardo, Maria Grazia De Angelis, and Marco Giacinti Baschetti.
2019. "Models for Facilitated Transport Membranes: A Review" *Membranes* 9, no. 2: 26.
https://doi.org/10.3390/membranes9020026