Thin-Film Composite Membrane Compaction: Exploring the Interplay among Support Compressive Modulus, Structural Characteristics, and Overall Transport Efficiency

Water scarcity has driven the demand for water production from unconventional sources and the reuse of industrial wastewater. Pressure-driven membranes, notably thin-film composite (TFC) membranes, stand as energy-efficient alternatives to the water scarcity challenge and various wastewater treatments. While pressure drives solvent movement, it concurrently triggers membrane compaction and flux deterioration. This necessitates a profound comprehension of the intricate interplay among compressive modulus, structural properties, and transport efficacy amid the compaction process. In this study, we present an all-encompassing compaction model for TFC membranes, applying authentic structural and mechanical variables, achieved by coupling viscoelasticity with Monte Carlo flux calculations based on the resistance-in-series model. Through validation against experimental data for multiple commercial membranes, we evaluated the influence of diverse physical parameters. We find that support polymers with a higher compressive modulus (lower compliance), supports with higher densities of “finger-like” pores, and “sponge-like” pores with optimum void fractions will be preferred to mitigate compaction. More importantly, we uncover a trade-off correlation between steady-state permeability and the modulus for identical support polymers displaying varying porosities. This model holds the potential as a valuable guide in shaping the design and optimization for further TFC applications and extending its utility to biological scaffolds and hydrogels with thin-film coatings in tissue engineering.


Model Development
2][3] Davenport et al. 2 prove that for the dense active layer during compaction, the active layer pore size stays almost the same before and after compaction, as evidenced by positron annihilation lifetime spectroscopy (PALS).Meanwhile, the support membrane undergoes pore collapse during this compaction process, as indicated by the liquid porosimetry and SEM. 2 Thus, if the relationship between membrane flux and the time-dependent support porosity is known, we can estimate the membrane flux.The sections below will provide a detailed derivation for the calculation of membrane-dependent flux.As shown in Figure S1a, in membrane compaction processes, the external hydraulic pressure difference across the membrane caused the compression of the support membranes, leading to the time-dependent change of the support membrane morphologies, and eventually resulting in the time-dependent change in the intrinsic permeability (and thus flux) of the support membrane.The initial morphology of the support membrane plays an essential role in the whole compaction process, as it not only sets up the initial conditions of the membrane compaction but also affects the subsequent time-dependent creep behavior of the support polymer.Therefore, the first task of compaction modeling is to establish a simplified model of support membrane with proper pore structures matching with SEM observations and realistic, measurable structural parameters.
The structural features of the flat sheet support membranes originated from the phase inversion process.Polymer solutions are cast on glass plates and then transferred to non-solvent such as water, resulting in the precipitation of the polymer and thus membrane formation.
Depending on the polymer precipitation rates, two distinct pore structures can form the "spongelike" pore morphology formed at slower solvent demixing rates, and the "finger-like" morphology formed at higher solvent demixing rates.These two morphologies are not mutually exclusive but co-exist in a support membrane, and their fractions are dependent on the preparation conditions. 6water and salt permeability in the "sponge-like" pores is normally smaller compared with the "finger-like" pores. 6In addition, membranes with fewer "finger-like" pores also have stronger mechanical properties.As a result, in the applications such as reverse osmosis or nanofiltration, the support membranes are dominated by the "sponge-like" pores, with much smaller fractions (< 0.2) of the "finger-like" region, 6 due to the demand of high solute rejections and higher transmembrane pressures, and these structures are also confirmed by SEM observations of commercial RO and NF membranes.
Therefore, in this work, we proposed to model the support membrane morphology as a uniform flat-sheet foam ("sponge-like") polymer with open cellular pores, at thickness (L).The porosity of the foam polymer (excluding any "finger-like" pores) is ϕf.The "finger-like" pores are modeled as cylindrical pores perpendicular to the membrane surface penetrating through the full thickness of the support foam polymer and are uniformly distributed.The locations of these terminologies are presented in Figure S1b.The fraction of all the cylindrical pores to the whole volume of the support is ϕcp.The radius of the cylindrical pores is r.Atop the support membrane is the active layer of the membrane, with thickness h.The following mathematical relationship can be derived (Equations 1-4).
The total mass of the solid polymer contained in the membrane (msp), the mass of the foam polymer (mfp), and the mass of the membrane should be the same (mm), and mass conservation is always valid during the whole compaction processes: Where ρ and V denote the corresponding density and the volume, respectively.According to the porosity defined above: Note that these equations are valid at all times during compaction, indicating that ϕf (t), ϕcp (t), Vsp(t), Vfp(t), and Vm(t) also obeys the above relationships at all t values.
In this work, we neglect all the shear stress among the active layer, support membrane, and the supporting mesh at the back of the support membrane.Only the normal stress induced by the transmembrane pressure exists within the supporting membrane.Upon compaction, the cellular foam polymer undergoes compressive strain in the same direction as the compressive stress.
Meanwhile, the cellular foam polymer expands laterally in the planar direction and occupies the volume of the cylindrical pores at a Poisson ratio ν.The real-time ρfp (t) can be derived in Equations 5 to 10: ()   (0) = ( + ∆) 2 ( + ∆)  2  (6) According to the definition of the Poisson's ratio, when the strain is not small and assuming a constant Poisson's ratio: Substituting equation 8 into 7 yields: Rearranging equation 5 and substituting equation 9 into equation 5, we have Dividing both sides by ρsp yields: 7   () Since ϕf (0) is a measurable quantity, substituting equation 11 into equation 3, the relationship between real-time porosity of the foam (ϕf (t)) and the strain of the foam (ε(t)) is revealed: The mean pore size, area pore density and the pore size distribution can be related to ϕcp (t) as follows: Acp is the total area of the cylindrical pores, Am is the membrane area.ri obeys a certain distribution type, such as normal distribution, gamma distribution, etc. n is the number of cylindrical pores.Assuming the number of cylindrical pores is constant during the compaction process, the reduction in the ϕcp with time will only result in a shrinkage of the pore radius ri.
Details of generating the membrane sample randomly distributed pores by using the above for calculation can be found in the supporting information.

Viscoelastic modeling of support membranes (Linkage arrow "Viscoelasticity" in Figure 1)
Upon compaction, the polymer exhibited time-dependent compressive strain, known as the viscoelastic behavior.The compressive strain leads to the change in the polymer morphology, such as the cellular pore deformation and the lateral expansion of the foam polymer (in the planar direction in Figure S1a), which makes porosities ϕf (t), ϕcp (t) as time-dependent functions.
Therefore, the purpose of this section is to develop a stress-strain-morphology relationship that can be used to derive the expressions of ϕf (t), and ϕcp (t) for the permeability calculations in Supporting Information section 1.3.
The compressive strain of a solid polymer (porosity equals zero), εpolymer (t), can be normalized by the compressive stress σ as a quantity called tensile creep compliance, Dsp(t): The transient tensile creep compliance of the solid polymer (Dsp) can be calculated by the combination of a Hookean spring, a Kelvin-Voigt element, and a dashpot: 8 where D0 is the compliance of the Hookean spring part, DKV represents the compliance of the Kelvin-Voigt element, τbur,0 is the retardation time, and η0 represents the viscosity of the material.
For membrane polymers, D0 is significantly smaller than Dkv and given by the time scale of membrane compaction (normally less than 10 5 s).Both the first term and the last term in equation 16 can be neglected.This indicates that a Kelvin-Voigt element is sufficient to capture the main feature of polymer creep behavior during membrane compaction: For open-cell foam polymers, previous works have shown that the modulus Efp of such foam materials can be related to the modulus of the solid polymer (Esp) as: Similarly, for the creep compliance of cellular foam polymer (Dfp), since Dfp is the inverse of Efp 9, 10 , now we have the compliances relationship between the foam polymer and solid polymer: where C1 is a constant with a scale of 10 0 .Hence we assumed C1=1 in our simulation.Substituting equation 12 into the above equation gives: From the definition of creep compliance, we have Substituting equation 21 to 20, we can now solve for the real-time strain of the cellular foam polymer by known quantities from equation 22.
Once () is known,   (),   () ,  () can be calculated accordingly using equation 19, 12 and 12, respectively, which will be useful for the next step to support permeability and S9 membrane flux calculations.The last step is to derive the expression of ϕcp (t).Recall equation 4: Rearranging equation 24 and substituting it into equation 23 gives: For the volume ratio of the membrane, since the area of the membrane is a constant, we have: Lm(0) is the membrane support thickness at the initial condition, and Lm(t) is the real-time thickness.εm (t) is the real-time strain of the membrane, and Dm (t) is the overall compliance of the membrane.Handge 8 derived the relationship between Dm (t) and Dcp (t) as in equation 27, assuming the transport path (cylindrical macrovoid pores) weakens the modulus of the membrane linearly: Where τm is the tortuosity of the support membrane.In this paper, we assume the tortuosity is only contributed by the macro scale cylindrical pores.If the pores are straight, τm = 1 and Eq.27 simplifies to Dm (t) =Dfp (t).Also, εm(t) =ε(t).
For the volume ratio of the cellular foam polymer ϕcp, we previously calculated it in equation 25.Substituting equation 26 and 27 into equation 25, eventually it gives: Note that σ here denotes the compressive stress inside the foam polymer membranes, but not the transmembrane pressure, ∆P.If ϕcp is small enough, σ ≈ ∆P, where ∆P is the transmembrane pressure.If ϕcp is not small, its relationship with ∆P can be derived from the force balance analysis as σ = ∆P/(1 -ϕcp).The influence of different support pore tortuosity under TFC membrane compaction by a transmembrane pressure Δp is shown in Figures S2a and S2b. 8 For TFC membranes, as the transmembrane pressure (TMP) is ∆p, the top surface of the whole TFC 28 has now become an implicit equation.Thus, we used the iterative method to solve for ϕcp (t) and ϕf (t), using the value from the previous step as the initial value.

Support membrane permeability calculations
Once the time-dependent porosities ϕf (t) and ϕcp (t) are known, the permeabilities of water can be calculated.The permeability for a specific cylindrical pore i with pore radius ri may be estimated by the Hagen Poiseuille equation: The permeabilities of the cellular foam polymer can be calculated from the Brace's equation: 11   () =     2   () 3 (30) Where dfp is the average cell diameter, and Afp is an empirical constant.In practice, dfp can be determined from the porosimetry measurements, and the coefficient Afp can be calculated by the measurement using a pure foam polymer.Both the scales of the Pcpi (t) and Pfp (t) are significantly larger than the permeability of the polyamide layer (Pa).

Water and solute flux calculations by resistance-in-series modeling and Monte Carlo simulations (Linkage arrow "Monte Carlo" in Figure 1)
Once the membrane morphology and the permeability of different membrane parts are known, the next step is to evaluate the water flux of the membrane.Water molecules must transport across the active layer of the membrane first, and then the path through either the foam polymer part, the cylindrical pore part, or both parts, before reaching the permeate side.The resistance-inseries approach treats one transport media with uniform transport property (e.g., water permeability, heat conductivity) as one resistor, and the total transport resistance as the summation of each transport resistance along a transport pathway.Thus, for a given water probe I placed on the surface of the membrane, its path-dependent transport resistance can be calculated by knowing the transport resistance within each part of the membrane.The resistance is determined by the pathway length and the permeability of each part of the membrane.In this work, the following three parts of the membrane are involved: 1. the transport pathway in the active layer, with a length of hAi; 2. the transport pathway inside the foam polymer of the support membrane, with a length of hfi; 3. the transport pathway in the cylindrical pores of the support membrane, with a length of hcpi.
the water flux of all the water probes and dividing the number of points used.Figure S3b shows the 3D model of one period cell with one support pore (pore size r1) in the center of the interface between the active layer and the support layer surface.One sample water probe i enters from the top surface of the active layer.should be used finally: For the calculation of the overall membrane water flux, we used large numbers of evenly distributed water probes on the periodic box to better estimate the water flux with respect to support membrane morphologies.The water flux of the membrane (JW) is then determined by the average water flux of all the water probes.
Thus, the real-time membrane flux can be solved by using the time-dependent morphology correlations derived in Equation 28.

Model validation and simulation Parameters
The simulation parameters are tabulated in Table S1, including the references.In this model, Dkv and τbur,0 is selected as the fitting parameters for different types of membranes when validating the model with experimental data.Other parameters are fixed parameters.Table S2 lists the relationship between the number of Monte Carlo simulation points and the calculation accuracy for the steady-state permeance.

Permeances and Permeabilities
The permeance of the active layer The permeability coefficient of foam polymer Table S2.Required numbers of evenly distributed Monte Carlo (MC) probes to accurately assess the relative permeance of the periodic cell with the cylindrical pore at the center in Figure S3.The steady-state relative permeances are values calculated using the baseline values listed in Table S1.
As the number of the MC probes increases, the relative permeance converges.Here, we set the relative permeance values when using 1×10  S3.Pore diameter ranges, calculation formulars and possible water permeability ranges for cylindrical pores, foam polymer and active layer.
Table S3 shows the possible permeabilities of the cylindrical pores (Pcp), foam polymer (Pf) and active layer (Pa).Based on the characterization techniques regarding their pore sizes, we can now safely argue that in TFC membranes, Pcp >> Pf >> Pa.Based on the above permeability sequence, we evaluated the minimum resistance transport pathway (MRTP) distributions in Figure S4.We find that even under extreme cases, when Pf = 10 Pa, the MRTPs are still located in the "interface layer" atop the foam polymer, which has a thickness comparable to the active layer but significantly thinner than the thickness of the support (Figure S4e).When Pf increases the thickness of this layer decreases.(Figure S4f)    S1.
membrane will be compressed by a force of ∆p*A, where A is the membrane area.Different from the polymer support alone, for example, UF membrane in Figure Error!Reference source not found.c,the transmembrane pressure ΔP = Δp on the UF membrane surface.For the support membrane of the TFC membrane, if the total area of TFC membrane surface is A, then the area of surface support will be A*(1-ϕcp).To maintain force balance, ΔP*A (1-ϕcp) = Δp *A, thus, the pressure exerted on the support membrane polymer (ΔP) will be Δp/ (1-ϕcp).An illustration is provided inError!Reference source not found.d.By incorporating the above expression, Equation

Figure S3 .
Figure S3.Schematic of the geometry used in the water transport path calculation.(a) TFC membrane surface top view.The dashed lines show the edges of four sample squares with four pores (random pore sizes r 1 , r 2 , r 3 , and r 4 ) located in the center of each sample square.Water probes that fall within the solid square area (for example, Probe P) need to calculate their shortest water transport pathways (δ 1 , δ 2 , δ 3 , and δ 4 ) with respect to the 4 pores located at the vertex of the solid square.(b) 3D model of the periodic pore cell with one cylindrical pore in the center with pore size r 1 from (a).One sample water probe P enters from the top surface of the active layer.(c)Cross-sectional view of (b).The determination of MRTP of probe P if P is placed atop the foam polymer (d) Comparison of MRTPs with the straight penetration scenario.Since Pcp >> Pf, and L >> h, the straightly penetrated situation is not common unless P is very far away from the cylindrical pores.

Figure S4 .
Figure S4.Minimum resistance transport pathways (MRTPs) distributions.(a) TFC membrane side-view structure.(b) TFC membrane side-view near the active layer -support membrane interface.(c) MRTPs locations if Pcp >>Pa = Pf.The MRTPs will always be in the active layer and the cylindrical pores.(d) MRTPs locations if Pcp = Pf >> Pa.The MRTPs will always be in the active layer and the cylindrical pores.(e) The realistic permeability scenario in TFC membranes, where Pcp >>Pf >>Pa.Here we assume Pcp = 10 7 Pf = 10 8 Pa, which might be the most extreme scenario in a TFC membrane since Pf is only 10 times the value of Pa.It seems that all the MRTPs for different Monte Carlo probes are still located in the thin layer at the top part of the active layer, with thickness comparable to the active layer thickness, which is referred to as the "interface layer" in this work (f) The effect of foam polymer permeability on the MRTPs distributions.It seems that the increase in Pf will lead to the MRTPs moving toward the interface of the active layer and the support foam polymer.These results highlighted the importance of the "interface layer" to membrane transport and their vital roles in membrane compaction.

Figure S6 . 8 Pa - 1 ,
Figure S6.Comparison of model with cylindrical pores and without cylindrical pores (foamonly support), assuming they are made of the same polymer foam.(a) Comparison of permeance decline of cylindrical pore plus foam polymer (C+F, in solid lines) and foam support (F, in dashed lines).(b) Comparison of the compressive strain of cylindrical pore plus foam polymer (C+F, in solid lines) and foam support (F, in dashed lines).It seems that the strains are similar with or without the cylindrical pores.(c) Flux decline fitting of a representative TFC membrane, NF-270, at 15 bar transmembrane pressure.Data source from Semião et al. 5 The fitting parameters are D kv = 5.0×10 -8 Pa -1 , τ 0 = 10 min.

Figure S8 .
Figure S8.Effect of initial foam porosity on the steady-state strain of the membrane.Apparently, the membranes with higher ϕ f (0) values are more vulnerable to compaction, but they experience less flux decline in Fig. 4g.

Figure S9 .
Figure S9.Effect of transmembrane pressure on the optimum ϕ f (0) of the membrane.

Figure S10 .Figure S11 .
Figure S10.Effect of initial foam porosity on the initial permeance (t = 0) and steady-state permeance of TFC membrane with foam-only supports.(Compare with Figure 4g)

Figure S12 .
Figure S12.The trade-offs between steady-state modulus and transport properties for TFC membranes with foam-only support polymers.The values in the legends represents the creep compliances in the units of Pa -1 .

Table S1 .
All parameters used in fitting NF and RO membrane flux vs time data in Figure 3a & 3b, and the baseline values used for the subsequent compaction modeling.Parameters marked with a red asterisk (*) represent fitting parameters.

Discussions regarding the distribution of the minimum resistance transport pathways (MRTPs) in this work Table
6numbers of MC points as the accurate value and calculated the percentage errors of using fewer MC points.Using 10000 MC points per unit cell could effectively control the percentage error of the relative permeance below 0.23%, and therefore, 10000 MC points were selected to evaluate the relative permeance in this work.(Marked as the green bold text in the table)