Nonlinear Flux–Pressure Behavior of Solvent Permeation through a Hydrophobic Nanofiltration Membrane

Abstract

Nonpolar solvents have been reported to exhibit a nonlinear flux–pressure behavior in hydrophobic membranes. This study explored the flux–pressure relationship of six nonpolar solvents in a lab-cast hydrophobic poly(dimethylsiloxane) (PDMS) membrane and integrated the permeance behavior in the evaluation of the proposed transport model. The solvents exhibited a nonlinear relationship with the applied pressure, along with the point of permeance transition (1.5–2.5 MPa), identified as the critical pressure corresponding to membrane compaction. Two classical transport models, the pore-flow model and solution-diffusion model, were evaluated for the prediction of permeance. The solution-diffusion model indicated a high correlation with the experimental results before the point of transition (R² = 0.97). After the point of transition, the compaction factor (due to membrane compaction after the critical pressure) derived from the permeance characteristics was included, which significantly improved the predictability of the solution-diffusion model (R² = 0.91). A nonlinear flux–pressure behavior was also observed in hexane–oil miscella (a two-component system), confirming the existence of a similar phenomenon. The study revealed that a solution-diffusion model with appropriate inclusion of compaction factor could be used as a prediction tool for solvent permeance over a wide range of applied transmembrane pressures (0–4 MPa) in solvent-resistant nanofiltration (SRNF) membranes.

1. Introduction

Membrane processing is gaining importance in the chemical processing industries. The increased use of solvents in a variety of industrial processes, including the food and pharmaceutical industries, and the drawbacks associated with the solvent recovery have stimulated the development of new separation techniques. Edible oil processing is the third-largest food-processing sector in the world; ∼73 MMT of vegetable oil is produced through solvent extraction,¹ which consumes a relatively large amount of energy for solvent recovery in the oil extraction process. Organic solvent nanofiltration (OSN) is a potential alternative to conventional evaporation,²⁻⁴ which focuses on separating nonaqueous systems without a phase change by employing solvent-resistant nanofiltration (SRNF) membranes. Therefore, it is imperative to understand the transport mechanism of solvents in SRNF membranes to predict their performance and for induction in the intended process step.

The fundamentals of aqueous NF are well established in the literature, while the transport mechanism for nonaqueous systems is not yet completely understood.^5,6 Several transport models have been proposed in the literature, most of which are adapted from the aqueous NF system. However, this adaption may not always be appropriate. When extending the principles from water to organic solvents, the physicochemical properties of the solvent and its interaction with the membrane significantly affect the mass transport through the NF membrane.³ The solvent–membrane interaction plays a crucial role in the transport of the solvent in the membrane⁷ and hence necessitates the need to understand the mechanism of the solvent’s interaction with the membrane, which can be vital in the rational design of new membrane–solvent combinations.⁸ It is observed that the transport of a nonpolar solvent is generally higher than that of a polar solvent in a hydrophobic membrane, which is attributed to the affinity of nonpolar solvents with hydrophobic materials like the PDMS membrane, as indicated by the higher contact angle values.⁹ For the transport mechanism of solvent permeation in SRNF membranes, primarily two types of mechanisms, namely, solution−diffusion^10,11 and pore-flow,¹² have been discussed and reported in the literature. Transport mechanisms with the combined characteristics of pore-flow and solution−diffusion have also been proposed.^13,14 Although the above attempts have contributed to the understanding, continued efforts are required to establish the transport mechanism in SRNF membranes.

For the evaluation of pore-flow and solution-diffusion models, the flux–pressure characteristics were estimated using solvent permeance (flux normalization with pressure). Increase in both linear and nonlinear pure solvent flux with increasing transmembrane pressure has been observed by various research groups (Table 1). The permeance (flux/pressure) has been often calculated at a single pressure point.^13,15 Membrane manufacturers recommend a higher pressure with SRNF membranes (of up to 3.5 MPa or even higher) to obtain a reasonable process flux. Accordingly, a wider experimental range of applied pressure is desirable while characterizing the membrane.

Table 1. Nonlinear Flux–Pressure Relationship Reported in the Literature for Solvent Transport in Hydrophobic Membranes.

s. no.	membrane description	solvents	pressure (MPa)	preconditioning	salient relevant observations	refs
1	PDMS-PAN (MPF-50, Koch)	decane, butyl acetate, acetone, pentanol, butanol, propanol, ethanol, methanol	0.5–3.0	performed an acetone run before each solvent run to obtain the standard solvent flux.	reported a linear effect of pressure on flux with nonpolar solvents and a nonlinear effect in polar solvents, attributed to the membrane compression. Identified viscosity and surface tension as the solvent properties influencing transport.	Machado et al.15
2	composite dimethyl silicone (membrane-D, osmonics)	ethanol, methanol isopropanol, octane, pentane, hexane	0.5–5.5	performed a polarity conditioning with the appropriate solvent/solvent mixtures before each solvent run.	observed a linear effect of pressure on the flux for both polar and nonpolar solvents.	Bhanushali et al.16
3	PDMS-PAN (GKSS)	n-hexane, n-heptane i-hexane, i-heptane i-octane, cyclohexane Xylene	0–0.9	performed an n-heptane run before each solvent run to obtain standard solvent flux.	reported a nonlinear effect of pressure on flux, attributed to either membrane compaction or the combined (viscous and diffusive) transport phenomenon.	Robinson et al.12
4	PDMS-PAN composite (GMT-oNF-1, Borsig)	1-dodecene	0.1–3.5	no pretreatment.	reported a nonlinear effect of pressure on flux and supported with membrane compaction visualization by scanning electron microscopy (SEM).	Zedel et al.17
5	PDMS-PAN composite (GMT-oNF-1, GMT-oNF-2, Borsig; HZG69, HZG70, Helmholtz-Zentrum, Geesthacht)	1-dodecene	0.1–3.5	no pretreatment.	observed a nonlinear effect of pressure on flux in all of the membranes and accounted it in terms of the effective membrane thickness in the solution-diffusion model.	Zedel et al.18
6	PDMS-PVDF composite (CSIR-CSMCRI)	cyclohexane, hexane hexadecane, petroleum ether, toluene, xylene	0.25–4.0	soaked in the respective solvent for 12 h before every experimental run.	observed a nonlinear effect of pressure on flux and identified a transition point on the flux–pressure characteristic curve corresponding to the critical pressure at which membrane compaction probably occurs.	this work

Taking inferences from the classical pore-flow and solution-diffusion models, the critical parameters were identified to develop a transport model that described the transport of nonpolar solvent permeation through the SRNF membrane. The present study deals with the detailed evaluation of the flux and pressure relationship of pure solvents as well as hexane–oil miscella in the lab-cast hydrophobic (PDMS) membrane, as it is pertinent to integrate permeance behavior for the absolute validity of the transport model evaluated over a wide pressure range (up to 4 MPa). The model proposed in the study was also used for predicting the permeate flux.

2. Theoretical Considerations

2.1. Nonlinear Flux–Pressure Relationship of Solvents

The linear and nonlinear flux–pressure relationship of polar and nonpolar solvents in membranes has been studied by various researchers (Table 1). In these studies, the viscosity and surface tension of the solvent, and the ratio of surface tension of the solvent to that of the membrane were stated to be the factors affecting the solvent transport in the membrane.

Robinson et al.¹² attributed the nonlinear behavior to either membrane compaction or a combined viscous and diffusive transport phenomenon. Zedel et al.¹⁷ explained the deviation by solvent-induced swelling of the dense active layer followed by a pressure-dependent compression, taking inferences from SEM images. The present authors explain this nonlinear behavior as follows. Swelling of the membrane occurs upon exposure to solvent, generally leading to higher permeance.¹⁹ However, when the applied pressure attains a critical value, the membrane suffers compaction, affecting the permeance. The conceptualization of membrane swelling followed by pressure-induced membrane compaction is illustrated in Figure 1.

Figure 1.

Typical nonlinear flux–pressure relationship of the solvent in a PDMS membrane.

This study focuses on the nonlinear flux–pressure relationship of solvents in the hydrophobic membrane by dividing the characteristic curve into two distinct zones: swelled and compacted zones. The swelled zone is defined as the linear zone of the curve passing through the origin, and the compacted zone is defined as the linear zone after the point of transition. The point of transition is calculated with respect to the change in the angle between the linear lines demarcating the swelled and compacted zones. The formula used for calculating the angle is as follows:

1

where A₁ and A₂ are the slopes between the flux and applied pressure in the swelled and compacted zones, respectively (Figure 1). The intersecting point where the angle (acute angle) between the two zones is maximum is considered as the point of transition, the corresponding pressure is referred to as critical pressure (P_c), and the corresponding slopes (A₁ and A₂) are denoted as permeances, A_1c and A_2c, respectively. For defining the transport after the point of transition, a compaction factor has been proposed in the study as described in the following equation.

2

2.2. Transport Models

Two mathematical models describing the liquid transport through a membrane as a function of the physicochemical and membrane properties are pore-flow and solution-diffusion models.²⁰ In a solvent system, the permeance characteristics (flux/pressure) change over the range of applied pressure. The rate of increase of flux with an increase in pressure remains constant up to a certain point (critical pressure, swelled zone), and beyond that, it reduces with the applied pressure (compacted zone). This behavior may be due to the solvent-induced swelling of the dense active layer followed by pressure-dependent membrane compaction leading to increased membrane density. Accordingly, the pore-flow and solution-diffusion models have been modified to account for the flux–pressure characteristics of the solvents.

2.2.1. Modified Pore-Flow Model

The pore-flow mechanism assumes that the transport of a liquid through a membrane takes place through the permeation pores.²¹ The pore-flow model also considers that the constant chemical potential gradient is derived from a pressure difference across the membrane that leads to convective flow through the pores in the membrane and is governed by the classical Hagen–Poiseuille equation¹² (eq 3).

3

The flux calculation using the modified pore-flow model for the nonlinear flux–pressure relationship of nonpolar solvents in the hydrophobic membrane could be described as follows

If P_i ≤ P_c, then the flux at P_i is calculated as (swelled zone, Figure 1)

4

If Pi > P_c, then the flux at P_i is calculated as (compacted zone, Figure 1)

5

where J_i is the flux at any pressure P_i, α and β are the grouped parameters of the swelled zone and compacted zone, respectively, μ is the solvent viscosity, λ is the compaction factor, and P_c is the pressure corresponding to the point of transition. Membrane structural features include surface porosity (ε), pore diameter (d), membrane thickness (l), and pore tortuosity (τ). For dense NF membranes, it is difficult to quantify the membrane structural features. Therefore, the flux of the solvent (J) is considered proportional to the driving force across the membrane (ΔP) and inversely proportional to its viscosity (μ), while the rest of the parameters (ε, d, l, τ) are assumed to be constant for a given membrane.¹² It indicates that for a given membrane at a constant temperature, the viscosity of the solvent is the only parameter that influences its permeance.

The Hagen–Poiseuille equation was found to be suitable only for a higher pore size (70 nm) inorganic membrane with alcohols²² and for a dense PDMS NF membrane with aliphatic and aromatic solvents.¹²

2.2.2. Modified Solution-Diffusion Model

The solution-diffusion model states that permeants dissolve in the membrane matrix and diffuse through the membrane.^10,11 Typically, the flux across the reverse osmosis membrane has been described in terms of pressure and concentration differences. Accordingly, the flux performance in the simplified solution-diffusion model for water is given by eq 6.

6

The compaction factor can be applied to the solution-diffusion model as described below.

If P_i ≤ P_c, then the flux at P_i is calculated as

7

If P_i > P_c, then the flux at P_i is calculated as

8

where J_i is the flux at any pressure P_i, D is the diffusion coefficient, K is the sorption coefficient, c₀ is the concentration of the permeant at the feed side, ν is the molar volume of the permeant, l is the membrane thickness, R is the ideal gas constant, T is the temperature, ΔP is the transmembrane pressure, ΔΠ is the osmotic pressure, P_c is the critical pressure, γ and δ are the grouped parameters of the swelled zone and compacted zone, respectively, and λ is the compaction factor.

The modified solution-diffusion model has been applied for determining the solvent flux in nonaqueous systems.^7,16,18

3. Results and Discussion

3.1. Nonlinear Flux–Pressure Relationship

3.1.1. Nonlinear Flux–Pressure Relationship for a Single-Component (Solvent) System

The present work explored the flux–pressure relationship of six nonpolar solvents by employing a PDMS hydrophobic lab-cast membrane over a pressure range up to 4 MPa. A nonlinear flux–pressure relationship was observed for all of the six nonpolar solvents with varying transition points (Table 2, Supporting Information Figure S1), and the typical relationship obtained with hexane is illustrated in Figure 2. Initially, the permeance (flux/pressure) was found to be constant up to a certain point (critical pressure), beyond which it reduced with applied pressure. The change in permeance was probably due to the pressure-induced change in the membrane structure.

Table 2. Permeances and Compaction Factor of the Solvents.

		permeance (L·m–2·h–1·MPa–1)
solvents	critical pressure (MPa)	swelled zone (A1c)	compacted zone (A2c)	compaction factor (λ = A2c/A1c)
hexane	1.5	24.4	8.5	0.34
petroleum ether	2.0	16.0	5.9	0.37
toluene	2.0	16.5	3.5	0.21
xylene	2.5	10.3	4.0	0.38
cyclohexane	2.0	10.9	4.4	0.40
hexadecane	2.0	4.2	3.4	0.83

Figure 2.

Effect of transmembrane pressure on the hexane flux in a lab-cast membrane and commercial membrane (GMT-oNF-1).

Similarly, a nonlinear behavior has been reported in PDMS membranes processing polar and nonpolar solvents (Table 1). Although the nonlinear behavior has been reported in many studies,^12,17,18 it was not explored for permeance calculations. In the present work, an analysis has been made regarding the nonlinear relationship reported with nonpolar solvents and PDMS membranes. Interestingly, a transition point could be identified in the nonlinear relationship between 1 and 2 MPa with 1-dodecene in all of the four different PDMS membranes employed in the study,^17,18 similar to the observation made in the present study demarcating swelled and compacted zones of the membrane with respect to applied pressure. Further, this transition is more evident in the case of high-flux membranes. Besides, the critical pressure increased with increase in solvent temperature in the range considered in the study (10, 20, and 45 °C). Robinson et al.¹² reported the nonlinear behavior with seven nonpolar solvents in a PDMS membrane. However, the transition seemed to happen at a much lower applied pressure (0.2–0.4 MPa) in the GKSS membrane used in the study.

The deviation in the flux behavior was attributed to the solvent-induced swelling of the polymer in the active layer followed by pressure-dependent compaction.^12,17 The potential explanation for the decreased permeance could be the membrane compaction. It could be inferred that change in the membrane structure would occur at the critical pressure, resulting in the transition of permeance characteristics. The flux was linear and passing through the origin till the point of transition (critical pressure,1.5–2.5 MPa; Table 2). In the rest of the pressure range (after the point of transition) also, the flux followed a linear relationship but with reduced permeance. Further, no significant change seemed to occur with the application of pressure. This linear deviation was also examined with a commercial membrane (GMT-oNF-1) and hexane, which revealed a similar trend (Figure 2).

In the present study, the swelling of the freestanding (PDMS) active layer was measured gravimetrically, and a significant swelling was observed with all of the solvents, as represented by sorption coefficient ranging from 0.53 to 2.22 g·g^–1 (Table 4). The swelling is schematically represented in the composite membranes in Figure 1. Further, it could be perceived that increase in applied pressure could result in the compaction of both active and support layers, in turn increasing the overall membrane resistance.

Table 4. Physical and Transport Properties of Nonpolar Solvents.

solvent	viscositya	molecular weighta	densitya	diffusion coefficientb	sorption coefficientc
	μ (mPa·s)	M (g·mol–1)	ρ (g·cm–3)	D (×109) (m2·s–1)	K (g·g–1)
hexane	0.31	86.18	0.66	1.95	2.06
petroleum etherd	0.46	82.20	0.64	1.11	2.09
toluene	0.56	92.14	0.87	1.12	2.18
xylene	0.70	106.17	0.87	0.86	1.97
cyclohexane	0.89	84.16	0.78	0.70	2.22
hexadecane	3.03	226.44	0.77	0.84	0.53

^aSource: Lide.²⁴

^bCalculated using Stokes–Einstein equation .

^cGravimetrically estimated from sorption studies.

^dPetroleum ether: mixture of pentane, 2-methyl pentane, 1-pentene, and cyclopentane.

In order to investigate the nature of compaction, further experiments were conducted with a lab-cast membrane and hexane in a cycle of increasing and decreasing applied pressures. There was no significant difference observed between the flux data measured at the corresponding set of experimental points (Figure 3), which indicated that there is no permanent physical effect of pressurization on the membrane. In other words, the effect of pressure was reversible and membrane compaction disappeared on the removal of applied pressure. SEM studies of swelled and compacted membranes also revealed that there is no permanent physical change in the membrane, corroborating the fact that the effect of pressure is reversible. From these results, it can be inferred that the swelling of the active layer and applied pressure-induced nonpermanent compaction influence the permeance performance of the solvent in the membrane used in the study.

Figure 3.

Effect of increasing and decreasing transmembrane pressure on the hexane flux in a lab-cast membrane.

3.1.2. Nonlinear Flux–Pressure Relationship for a Two-Component (Solvent–Oil) System

Operating pressure is a determining factor for any economical process design. Therefore, the flux–pressure relationship trend was verified with a two-component system consisting of a solvent and a solute (25% hexane–oil miscella). The effect of applied pressure on the hexane flux and oil flux in the hexane–oil miscella system is illustrated in Figure 4. The inclusion of solute (oil) reduced the hexane flux (5 L·m^–2·h^–1) in the miscella system by nearly one order of magnitude when compared to the pure solvent flux (51 L·m^–2·h^–1 at 3.5 MPa). At the same time, the oil flux (0.4 L·m^–2·h^–1) increased by two-fold compared to the undiluted system (0.2 L·m^–2·h^–1), probably due to the oil–hexane coupling effect. However, the permeates, solvent and solute, both displayed a nonlinear behavior similar to the single-component system in the pressure range of 0.25–3.5 MPa. The transition of permeance was observed for both the components after the critical pressure. A higher applied pressure on the membrane surface may lead to pore contraction (loss of porosity) and a decrease in the free available volume in all layers of the membrane to some extent, leading to a significant increase of the membrane layer resistance.¹¹

Figure 4.

Nonlinear flux–pressure relationship of the solvent and oil in a lab-cast membrane (pure solvent flux at 3.5 MPa – 51 L·m^–2·h^–1; oil flux at 3.5 MPa – 0.2 L·m^–2·h^–1).

3.2. Transport Model for the Two Linear Zones

Two transport models, namely, pore-flow and solution-diffusion models, describing the solvent transport through an SRNF were evaluated. The transport parameters were identified as the viscosity for the pore-flow model, and the product of the diffusion coefficient and sorption coefficient for the solution-diffusion model. For model verification, the influence of the respective parameter was tested with the permeance of the solvent, which was determined as the slope of the permeate flux and transmembrane pressure. Taking the flux–pressure characteristics into consideration, the permeance calculation for six nonpolar solvents was carried out in two parts: the permeance of the solvents before the critical pressure (first linear swelled zone) and after the critical pressure (second linear compacted zone). The permeance of the first and second linear zones was determined as the slope of flux versus pressure.

3.2.1. Transport Model before the Point of Transition (Swelled Zone)

The Hagen–Poiseuille equation is synonymous with the pore-flow model.¹² The relationship between the permeance of solvents and their viscosity plays an important role,¹⁵ but it is rather inadequate in predicting the permeance of organic solvents in dense membranes.

According to the solution-diffusion model, the permeance is directly proportional to the product of diffusion coefficient and sorption coefficient of the solvent. A very high correlation coefficient (R² = 0.97) was observed between the permeance and the product of diffusion coefficient and sorption coefficient (solution-diffusion model), while the correlation was comparatively lower (R² = 0.92) in the case of permeance and the inverse of viscosity (pore-flow model) (Figure 5). Besides, solvent diffusivity in the membrane matrix is inversely proportional to the solvent viscosity according to the Stokes–Einstein equation (eq 9). Taking all this into consideration, it looks reasonable to conclude that the transport of solvents in dense membranes follows the solution-diffusion model.

Figure 5.

Influence of (a) viscosity (pore-flow model) and (b) diffusion coefficient and sorption coefficient (solution-diffusion model) on the flux of nonpolar solvents in the swelled zone. Permeance (flux/pressure) is calculated as the slope of a linear line passing through the origin [μ: viscosity; D: diffusion coefficient; K: sorption coefficient].

3.2.2. Transport Model after the Point of Transition (Compacted Zone)

The permeance after compaction was calculated using the point of transition as the origin in defining the linearity of the second zone. In this zone, both pore-flow and solution-diffusion models did not reveal a good fit. The permeance characteristic depicts two clear linear zones (Figure 2), and the transitional change in permeance with respect to pressure was examined as a reflection of compaction. Accordingly, a compaction factor (λ) has been proposed in this work as described in eq 2 to account for the membrane compaction (Table 2). This ratio can be used as a representation of the degree of compaction and has been factored as the decrease in porosity of the membrane in the assessment of the models.

Inclusion of compaction factor drastically improved the performance of the pore-flow model (R² = 0.83) and more so in the solution-diffusion model (R² = 0.91) (Figure 6; Table 3). The results suggest that the transport of the solvent after compaction also follows the solution-diffusion mechanism. The analysis has confirmed the phenomenon of compaction and the corrective role of compaction factor in the permeance calculation. Hence, it should be factored in the estimation of the permeance in the higher-pressure zone (after the point of transition), wherein often lies the practical range of applied pressure for the nonaqueous applications of dense NF membranes.

Figure 6.

Influence of (a) viscosity (pore-flow model) and (b) diffusion coefficient and sorption coefficient (solution-diffusion model) on the flux of nonpolar solvents in the compacted zone, where a₁, b₁ represent the models without the compaction factor and a₂, b₂ represent the models with the compaction factor. Permeance (flux/pressure) is calculated as the slope of a linear line with origin shifted to the point of transition [μ: viscosity; λ: compaction factor; D: diffusion coefficient; K: sorption coefficient].

Table 3. Transport Model Assessment for Swelled and Compacted Zones.

	swelled zone		compacted zone
			without correction		with inclusion of the compaction factor
model assessment parameters	pore-flow model	solution-diffusion model	pore-flow model	solution-diffusion model	pore-flow model	solution-diffusion model
R2	0.92	0.97	0.48	0.43	0.83	0.91
RMSE	1.74	1.07	1.29	1.35	0.74	0.54

3.3. Comparative Assessment of the Models

The pore-flow and simplified solution-diffusion models were evaluated for all of the six solvents taken up for the study. These models were fitted using linear regression, and the coefficient of determination and root-mean-square error (RMSE) were estimated (Table 3). Higher correlation coefficients, along with lower RMSE values, are the most desirable determinants for model assessment. Accordingly, the solution-diffusion model was found to be the best fit for nonpolar solvent transport in the hydrophobic membrane. The permeances before and after the point of transition can be sufficiently explained by the solution-diffusion model, which displayed high correlation and lower RMSE values of R² = 0.97, RMSE = 1.07 and R² = 0.91, RMSE = 0.54, respectively, with the predicted values (Table 3). The inclusion of the compaction factor estimated using permeance characteristics improved the model prediction and accounted well for the nonlinear flux–pressure behavior of solvents after the point of transition. Considering the model performance, it could be concluded that the solution-diffusion model, which describes the diffusional flow, is a better model for determining the permeance of nonpolar solvents in a dense hydrophobic membrane. An exemplary solution-model flux calculation for the nonlinear flux–pressure relationship has been illustrated in the Supporting Information File (Supporting Information Figure S2).

4. Conclusions

The transport phenomenon of nonpolar solvents through a hydrophobic membrane was developed after assessing the existing pore-flow and solution-diffusion models. The study encompasses the nonlinear behavior of the flux–pressure relationship by identifying a transition point demarcating the swelled and compacted zones of the membrane with respect to the applied pressure. This behavior has been appropriately synchronized in the transport model study. A compaction factor was deduced and appropriately incorporated into the solvent transport model for permeance after the point of transition, which accounts for the nonlinear behavior. The solution-diffusion model predicted well the permeance of nonpolar solvents in the hydrophobic membrane before and after the point of transition. The nonlinear flux–pressure behavior was also observed in hexane–oil miscella (a two-component system). Future work would focus on the relative permeance of the solvent and the solute in hydrophobic NF membranes towards applications related to desolventizing and deacidifying vegetable oil miscella. Furthermore, it would be interesting to study the existence of this nonlinear flux–pressure phenomenon in other polymeric membranes.

5. Materials and Methods

5.1. Materials

An SRNF membrane with organosilica functionalized PDMS (∼2.5 μm thickness) as the active layer and a mixed-matrix poly(vinylidene fluoride) (PVDF)–polyester composite film (∼25 μm thickness) as the support layer was prepared using the dip-coating technique²³ and supplied by CSIR-CSMCRI, Bhavnagar, India. The PDMS active layer (0.5–1 mm thickness) was also supplied along with the membrane for conducting sorption studies. The commercial PDMS composite membrane (GMT-oNF-1) was procured from BORSIG Membrantechnik GmbH, Berlin, Germany.

The stability of the membrane was evaluated by soaking the membrane in the solvent for 24 h, followed by visual examination. The membrane was cut into circular discs of 49 mm diameter (effective membrane area 14.6 cm²) for use in the membrane cell.

Analytical-grade solvents (hexane, petroleum ether, toluene, xylene, cyclohexane, hexadecane) were purchased from M/s Sisco Research Laboratories, Mumbai, India. The solvents selected for this study represent a wide range of nonpolar solvents varying in viscosity (0.31–3.03 mPa·s) (Table 4). Refined rice bran oil was purchased from the local market for the two-component study.

5.2. Experimental Setup and Permeation Runs

All of the permeation runs were conducted in a dead-end membrane filtration cell (Model: HP4750, Sterlitech, Kent) at a transmembrane pressure of up to 4 MPa and at room temperature (25 ± 2 °C). The membrane disc was positioned at the bottom of the filtration cell in such a way that the active layer comes in contact with the feed. The membrane cell was placed on the magnetic stirrer, and the spin bar speed was maintained at 600 rpm (Figure 7).

Figure 7.

Schematic diagram of the stirred membrane cell setup.

The membrane disc was replaced for each solvent study. The membranes were soaked in the respective solvent for 12 h of preconditioning before the experimental run. For each experimental run, 100 mL of solvent was charged into the cell, and the run was continued until it reached the desired limit of permeation (50%). In the case of hexane–oil miscella, 100 g of miscella was loaded in the cell and the permeation was limited to 10% to practically maintain the feed concentration. The membranes were stored in the solvent between the experimental runs. The permeate flux of each experimental run was always compared with the initial permeate flux of the respective solvent to ensure the solvent stability of the membrane.

Permeate flux values were obtained by calculating the volume of solvent permeating through the unit area of the membrane.²¹ Permeance is calculated as the slope of the permeate flux to the transmembrane pressure. A freshly cut membrane disc was used for each solvent. The same membrane disc was employed for performing triplicate runs at different experimental points within the specific solvent system. The permeance data presented are the average values.

5.3. Sorption Studies

Pre-weighed cut coupons (∼3 × 3 mm²) of freestanding dense, dry PDMS films were immersed in the solvent for 12 h. The method reported by Fang et al.²⁵ was suitably adopted for the pure solvent system. The swollen sample was removed, and the excess surface solvent was gently wiped with tissue paper and weighed immediately (∼15 s) to minimize the error from solvent evaporation. These measurements were made with three different PDMS coupons for each solvent to get a reasonable assessment. The sorption coefficient of the pure solvent was calculated as the ratio of the weight of solvent absorbed to the dry weight of the membrane.²⁶

5.4. Model Fitting and Statistical Analysis

Pore-flow and solution-diffusion models were fitted using linear regression on the grouped parameters of the swelled zone (α, γ) and compacted zone (β, δ) (pore-flow model: ε, d, l, τ; solution-diffusion model: c, v, l, R, T).

For the solution-diffusion model, the diffusion coefficient of the solvents was calculated using the Stokes–Einstein equation²⁷

9

where D is the diffusion coefficient, k is the Boltzmann constant, T is the temperature, μ is the dynamic viscosity, and r is the spherical radius of the diffusive molecule. It is challenging to measure solvent diffusion in membrane material, as it is dependent on many membrane characteristics like density, cross-linking degree, and the exact recipe and procedure of membrane production.¹⁸ Therefore, considering the difficulties involved in obtaining reliable values, bulk diffusivities of solvents have been used in the present study. The molecular radius was calculated from the fundamental relation²⁸ as

10

where ν is the molar volume, ρ is the density, and N_o is the Avogadro number.

The model values were compared statistically with the experimental flux data. The coefficient of determination (R²) and root-mean-square error (RMSE) were calculated using the formulae given below²⁹

11

12

where A_exp and A_pre are the experimental and predicted permeances, respectively, A_expmean is the mean of the experimental permeances, and n is the number of observations.

Glossary

Nomenclature

A_1c:

permeance before the point of transition (L·m^–2·h^–1·MPa^–1)

A_2c:

permeance after the point of transition (L·m^–2·h^–1·MPa^–1)

A_exp:

experimental permeance (L·m^–2·h^–1·MPa^–1)

A_pre:

predicted permeance (L·m^–2·h^–1·MPa^–1)

c_o:

concentration of permeant at the feed side (g·mol^–1)

d:

pore diameter (m)

D:

diffusion coefficient (m²·s^–1)

l:

membrane thickness (m)

J:

flux (L·m^–2·h^–1)

k:

Boltzmann’s constant (J·K^–1)

K:

sorption coefficient (g·g^–1)

M:

molecular weight (g·mol^–1)

A₁:

slope of flux–pressure relationship of the swelled zone (L·m^–2·h^–1·MPa^–1)

A₂:

slope of flux–pressure relationship of the compacted zone (L·m^–2·h^–1·MPa^–1)

n:

number of observations

N_o:

Avogadro number (mol^–1)

ΔP:

transmembrane pressure (MPa)

P_i:

pressure at i (MPa)

P_c:

pressure at the point of transition (MPa)

r:

spherical radius of the molecule (m)

R:

ideal gas constant (J·K^–1·mol^–1)

R²:

coefficient of determination

T:

temperature (K)

ν:

molar volume (m³·mol^–1)

Glossary

Greek symbols

α, β, γ, δ:

grouped parameters

λ:

compaction factor

ε:

surface porosity

μ:

viscosity (Pa·s)

η:

dynamic viscosity (Pa·s)

τ:

pore tortuosity factor

π:

osmotic pressure (Pa)

ρ:

density (g·cm^–3)

Glossary

Abbreviations

OSN:

organic solvent nanofiltration

SRNF:

solvent-resistant nanofiltration

MMT:

million metric tonnes

NF:

nanofiltration

PDMS:

poly(dimethylsiloxane)

PVDF:

poly(vinylidene fluoride)

RMSE:

root-mean-square error

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.1c03624.

• Figure S1: Effect of transmembrane pressure on the solvent flux in the PDMS lab-cast membrane; Figure S2: A sample calculation for the determination of flux at any applied pressure using the solution-diffusion model in the swelled and compacted zones (PDF)

The authors declare no competing financial interest.