Temporal fluctuations of non-equilibrium water accessible pathways in polyamide reverse osmosis active layers

Abstract

Molecular-scale simulations of pressure-driven transport through polyamide domains were performed. Analysis revealed the importance of non-equilibrium molecular distributions and the existence of connected pathways from feed to permeate at all times. Individual pathways were found to be ephemeral with an observed upper bound persistence time of 1.75 ns.

1. Introduction

With advances in both laboratory and computational approaches, the structure of reverse osmosis polyamide membranes has become more resolved and interpretable at the molecular scale. Historically, an adequate understanding of the membrane structure was lacking, and perhaps as a result, the two most common transport models, i.e., the solution–diffusion and pore flow models, depend on diametrically opposed assumptions regarding the structure of the polyamide active layer (Wijmans and Baker, 1995; Paul, 2004). Recent work has focused on moving beyond these overly simplified models, although a firm connection between molecular-scale observations and larger scale models has not been established (Wang et al., 2023, 2021; Oren and Biesheuvel, 2018; Zhu et al., 2024). Such an approach could underpin the development of predictive property–performance relationships, and aid in the design of more efficient membranes.

Molecular dynamics (MD) simulation of water permeation is a fundamental method to model transport processes and reveal the morphological and topological properties of polyamide-water systems (Ridgway et al., 2017; Thompson et al., 2022). The morphology and topology of the pore structure of a porous medium is known to be a leading-order determinant of the permeability (Celia et al., 1995; Valvatne and Blunt, 2004), and at the molecular scale, polyamide has been shown to be a porous medium (Kim et al., 2005; Cahill et al., 2008; Chen et al., 2011). MD studies of polyamide have included broad measures of structure such as pore size distributions (Araki et al., 2015; Zhang et al., 2019; Muscatello et al., 2017; Ding et al., 2014; Shen et al., 2016; Hughes and Gale, 2010; Wang et al., 2016), and recent works have introduced connectivity, a measure of topology (He et al., 2023; Qian et al., 2025). Reconstruction of the polyamide structure from MD simulations has thus far relied on equilibrium assumptions, despite transport through RO membranes being an inherently non-equilibrium process (He et al., 2023, 2024; Sarkisov et al., 2020). A rigorous quantification of the polyamide morphology and topology is necessary to determine mechanistic relations between polyamide structural properties and observed performance.

The overall goal of this work is to advance the fundamental understanding of how morphology and topology influence transport through polyamide active layers. The specific objectives of this work are to: (1) utilize non-equilibrium permeation results to reconstruct the porous structure of the polyamide; (2) assess the connectivity of the polyamide structure by examining whether water accessible pathways that span from feed to permeate boundaries exist within polyamide active layers; (3) determine the time persistence of these pathways; and (4) identify how these pathways are occupied with water. The results from this work provide necessary information for the development of next-generation transport models based on a molecular understanding of polyamide structure.

2. Methods

2.1. Molecular dynamics simulations

MD simulations of crosslinked aromatic polyamide were performed ab initio following the approach reported in Vickers et al. (2022) and detailed in the Supplementary Information (SI). The simulated domain was 175 × 175 × 200 ų and consisted of 90% crosslinked and hydrolyzed m-phenylenediamine (MPD) and trimesoyl chloride (TMC). Fig. 1(a) depicts the simulation domain, including feed, permeate, and hydrated polyamide (gray), with water molecules (H: red, O: white) confined by graphene pistons (cyan) along the $z$ direction and periodic boundary conditions in the $x$ and $y$ directions. The yellow box indicates the analyzed subdomain. To create a pressure difference during permeation simulations, a force of 100 atm was applied to the feed piston. The permeate reservoir was maintained via the permeate piston at 1 atm. After 10 ns of equilibration, permeation simulation was carried out for 140 ns with atom locations saved every 0.01 ns.

Fig. 1.

(a) Simulated polyamide domain; and (b) time and cross-section averaged densities along the direction of flow.

2.2. Non-equilibrium porous structure

Reconstruction of the porous structure of polyamide membranes from molecular dynamics (MD) simulations involves connecting the spatial scales between MD simulations and the larger, averaged scale that governs membrane performance, with measures such as pore extent, connectivity, tortuosity, and permeability. Approaches to connect these spatial scales have thus far relied on equilibrium assumptions (Wang et al., 2023; He et al., 2022, 2023; Sarkisov et al., 2020). Specifically, the van der Waals radius is used to reconstruct the polyamide structure and the Coulombic radius of water (1.4 Å) is used to determine the water accessible pore space. In this equilibrium approach, all polyamide atoms of the same element (C,H,N,O) are treated identically, thereby neglecting the effects of partial charges, orientation, and dynamic, non-equilibrium interactions.

Because water permeation through polyamide is fundamentally a non-equilibrium process, relaxation of the equilibrium assumption is necessary to adequately describe the porous structure of polyamide (Bhat and Purisima, 2006). To achieve this, in this work, all observed distances ($r$) between the oxygen atom of water molecules ($\mathrm{O}\text{-}{\mathrm{H}}_2\mathrm{O}$) and each type of constituent polyamide atom ($X$) are binned, categorizing the interactions based on atomic partial charge, resulting in 17 unique atom types (see SI). From the observed distances, the radial distribution function $g$ is computed as

$$g_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}(r)=\frac{d{N}_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}}{4\pi r^2{\rho }_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}dr},$$ (1)

where $d{N}_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}$ is the bin count and ${\rho }_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}$ is the number density for each atom type. From the radial distribution function, the potential of mean force $G$ for each polyamide atom is calculated as (Hamelryck et al., 2010; Sippl, 1990)

$$G_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}(r)=-k_{B}T\text{ln}{g}_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}(r),$$ (2)

where $k_b$ is the Boltzmann constant, and $T$ is the temperature. The potential of mean force provides a measure of the average free energy landscape experienced by a water molecule as a function of its distance from a given polyamide atom type, inherently accounting for the average relative orientation of the water molecules and polyamide atoms. These distances are not theoretical; rather, they correspond to observed distances in the simulation, each occurring with varying frequency. Inversion of $G_{\mathrm{O}-{\mathrm{H}}_2\mathrm{O},X}(r)$ , accomplished through linear interpolation, yields a corresponding cutoff radius ($r_X$) for each atom type based on a selected value of $G$. The cutoff radius delineates the space occupied by the solid polymer, as opposed to the water. Larger values of $G$ correspond to lower-probability interactions and smaller cutoff radii.

To generate the polyamide structure, the simulation domain is tessellated into voxels and a binary indicator function $I$ is introduced. For each voxel, with a center located at $\mathbf{x}$, its classification as either a water-accessible voxel ($I=1$) or a polyamide voxel ($I=0$) is determined by treating every polyamide atom of a given type ($n$) as a sphere with a center at ${\mathbf{x}}_{nX}$ and a radius of $r_X(G)$, with the following form (see SI)

$$I(\mathbf{x})=\{\begin{array}{cc}1\hfill & {\Vert \mathbf{x}-{\mathbf{x}}_{nX}\Vert }_2>r_X(G)\forall nX\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}\phantom{\}}$$ (3)

Selecting a particular value of $G$ enables reconstruction of the water-accessible porous structure of the polyamide, establishing a direct link between membrane morphology and the probability of non-equilibrium water–polymer interactions.

2.3. Pore morphology and topology

To examine the morphology and topology of the polyamide, the domain was discretized with a voxel resolution of 0.05 Å in every dimension, producing a domain with 49.5 billion voxels. Then a connected components analysis of the water accessible pore space was performed (see SI) (Kong and Rosenfeld, 1989). This analysis yielded sets of water accessible voxels that were examined to determine whether they were connected to the feed side, the permeate side, or both within the polyamide active layer. If a set of water accessible voxels was connected to both the feed and the permeate side of the domain, the set of voxels was designated as a water accessible pathway. A Euclidean distance transform was also applied to the water accessible pore space to examine pore sizes (Saito and Toriwaki, 1994).

For analysis of the polyamide porous structure, the porous media morphology and topology toolkit, PMMoTo (https://tmweigand.github.io/PMMoTo/), a scalable, efficient, and parallel toolkit, was used. All morphological analysis was performed on Bridges-2 at the Pittsburgh Supercomputing Center using 216 AMD EPYC 7742 cores.

3. Results and discussion

The water permeability computed from the simulated water flux was 1.48×10⁻⁶ Lm⁻¹hr⁻¹bar⁻¹, which is consistent with experimentally measured values ranging from 0.1–1.1×10⁻⁶ Lm⁻¹hr⁻¹bar⁻¹ (Lin et al., 2016, 2018; Yang et al., 2019; Vickers et al., 2022), which were estimated from a reported water permeance (Yang et al., 2019) by assuming an active layer thickness of 110 nm based on previous analysis of comparable active layers (Lin et al., 2018).

A dense polyamide region, $z\in [-35,65]$ Å, was observed consistent with a previous observation and believed to be a scale-dependent feature, Fig. 1(b) (He et al., 2023). Specifically, a similar dense region has only been reported in polyamide membranes with a thickness of at least 150 Å, and is thought to arise from depth-dependent swelling ratios (He et al., 2023). The total average density, calculated from $z\in [-100,100]$ Å, was found to be 1.23 g/cm³, which is in good agreement with experimental values (He et al., 2022). Additional bulk properties of the polyamide domain are included in the SI.

Figs. 2(a) and 2(b) depict the radial distribution function and potential of mean force between $\mathrm{O}-{\mathrm{H}}_2\mathrm{O}$ and O-CONH (see SI for additional atom types) for 14,000 time steps, corresponding to 140 ns of permeation with output every 0.01 ns. Over 136 billion interactions between $\mathrm{O}-{\mathrm{H}}_2\mathrm{O}$ atoms and all polyamide atom types were quantified by computing separation distances within a 7 Å radius (5× the Coulombic radius of water) centered on each $\mathrm{O}-{\mathrm{H}}_2\mathrm{O}$ atom for every time step. For each atom type, distances that are less than the assumed equilibrium cutoff radius were observed, depicted by the red vertical line in Figs. 2(a) and 2(b). The equilibrium cutoff radius does not always coincide with the minimum of the potential of mean force, and was observed to be both smaller and larger. This underscores the limitations of the equilibrium approach in accurately capturing the behavior of non-equilibrium systems. The maximum observed $G$ value was 17.32 kJ/mol, representing the highest energy threshold at which all polyamide atom types were still supported by observed distance to $\mathrm{O}-{\mathrm{H}}_2\mathrm{O}$ data.

Fig. 2.

(a) Radial distribution function between $\mathrm{O}-{\mathrm{H}}_2\mathrm{O}$ and O-CONH; (b) potential of mean force for same atom pair; and (c) an instance of the polyamide structure with $G=17.32$ kJ/mol.

Fig. 2(c) depicts an instance of the polyamide structure, the water accessible pore space with the commonly used equilibrium assumption, and the additional non-equilibrium water accessible pore space, where the maximum $G$ value of 17.32 kJ/mol was used to determine the cutoff radii for each of the 17 atom types (see SI). A portion of the domain in Fig. 2(c) was omitted to highlight the change in the morphological characteristics in the dense polyamide region trending towards smaller water accessible pore space. With the equilibrium assumption, the average water accessible volume fraction of the polyamide was found to be 0.29. At the maximum observed $G$, this fraction increases to 0.51, providing an upper bound estimate.

To assess the connectivity of polyamide structure and whether water-accessible pathways exist within the polyamide porous structure, an optimization was performed at each timestep to determine the minimum $G$ required for a continuous path to form from feed to permeate, noting that such a continuous pathway may not exist at any given $G$. After analysis of the 14,000 unique temporal snapshots of the polyamide, every instance contained a water accessible pathway. Fig. 3(a) shows the distribution of the minimum $G$ such that a water accessible feed-to-permeate pathway exists. The average value of the minimum $G$ was 1.40 kJ/mol and the largest was 3.61 kJ/mol. This relatively low average minimum potential of mean force needed to form a water accessible pathway from feed to permeate, compared to the maximum observed $G$ of 17.32 kJ/mol, suggests that feed-to-permeate connections are not rare events, but rather occur frequently within our dataset.

Fig. 3.

Potential of mean force analysis of connected paths: (a) minimum; (b) continuous water accessible path persistence; and (c) maximum pore radius for $z\in [-35,65]$ Å.

To examine the persistence of the feed-to-permeate pathways, or the duration of sequential snapshots with the same feed-to-permeate connection, a connected component analysis of the water accessible pore space was performed for selected values of $G$ at each time step. Fig. 3(b) depicts a histogram of the persistence times of feed to permeate connections. At low $G$ (e.g., 1.40 kJ/mol), connections are short-lived, typically lasting only < 0.01–0.02 ns, if they exist at all. In contrast, at the maximum observed $G$, connections persist for up to 1.73 ns, with an average of 0.48 ns. This indicates that while the pathways exist on timescales relevant to transport (Vickers et al., 2025), they remain ephemeral.

Fig. 3(c) shows the maximum pore radius as a function of $G$ within the dense region of the polyamide. The gradual increase in average maximum pore size as $G$ is increased suggests a relatively consistent structure, as no significantly larger pores are observed at high values of $G$. For the intermediate $G$ value, the largest pore was not found on a connected pathway 7% of the time, indicating that large isolated pores exist at a given instance. However, at the largest $G$, the largest pore was always found on the connected pore space.

Previous MD studies have not observed feed-to-permeate pathways (He et al., 2023). Others have posited transient (Wang et al., 2023) or continuous (Ridgway et al., 2017; Luo et al., 2011; Shen et al., 2016) connections. These differences may stem from equilibrium assumptions, limited domain sizes, or reliance on water trajectories alone, which do not explicitly account for polyamide atoms. Note that continuous feed-to-permeate pathways are not required for transport to occur through a time-varying porous structure; however, examination of these pathways provide bounded estimates of the behavior and bears upon the debates between the solution–diffusion and pore-flow models.

Fig. 4(a) shows a snapshot of a representative feed-to-permeate pathway, where the water accessible pore space in the dense polyamide region is highlighted in blue and both the polyamide and disconnected pore space are omitted for clarity. The water accessible pore space outside of the dense region (shown in gray) is a highly interconnected porous network, which is consistent with prior descriptions (He et al., 2023). However, a transition in the morphological characteristics appears within the dense polyamide region, resulting in areas of isolated connectivity. This shift suggests the presence of a distinct regime within the membrane interior that may restrict transport through the active layer. Fig. 4(b) quantifies the averaged pore space and water volume fractions along the primary direction of flow for the entire pore space and for the feed-to-permeate connected pore space with $G=3.61$ kJ/mol. In the dense polyamide region, the connected pore space accounts for 18% of the total volume (69% of the total pore space) and 74% of the water is found in the connected pore space. At the minimum $G$, the total pore space fraction was 17% but no connected path existed. For the largest $G$, the connected pore space accounts for 35% of the total volume (83% of the total pore space) and 93% of the water is on a connected path. The choice of $G$ strongly affects conclusions about connectivity and volume fractions: while the minimum $G$ suggests no connected pathways in this snapshot, higher values reveal substantial connected pore networks. Outside of the dense region, the connected water and pore space fractions are roughly equal at all values of $G$, highlighting the importance of the dense region as a bottleneck to water permeation.

Fig. 4.

(a) Porous structure of a connected path; and (b) volume fraction of the pore space and water along the direction of flow.

Recent investigations into transport mechanisms through polyamide channels have revealed that water transport through polyamide spans a spectrum from sub-diffusive to diffusive behavior (Vickers et al., 2025; Mehlis and Wessling, 2025). However, this does not imply that the solution–diffusion model is the most appropriate framework as uniform diffusion through the membrane will not occur due to the presence of ephemeral pathways. Similarly, the presence of permeate-to-feed pathways within the membrane does not inherently validate the pore flow model as connected pathways are transient and evidence is available supporting diffusive transport (Vickers et al., 2025; Mehlis and Wessling, 2025). Instead, these findings should guide membrane design by examining if and why specific regions of the polyamide allow water to transport more readily. The findings should also inform the development of more accurate transport models that prioritize translating molecular-scale phenomena into larger-scale property–performance relationships and account for the dynamic, heterogeneous porous structure of the polyamide.

4. Conclusions

Analysis of molecular dynamics simulations revealed that the commonly used equilibrium assumption for analyzing pore morphology and topology is inconsistent with the inherently non-equilibrium, pressure-driven transport observed in polyamide membranes. To address this, a new approach was developed to link non-equilibrium transport behavior to the underlying porous structure. Using this approach it was shown that feed-to-permeate pathways are not rare events, with an average minimum required potential of mean force of 1.4 kJ/mol, and existed at every timestep of the simulation. Upper bound estimates indicate that these pathways may persist for up to 1.73 ns, but they are ephemeral. These insights offer a molecular-level understanding of transport that can inform the development of membrane property and performance relations by accurately reconstructing the porous structure of the polyamide through direct linkage to molecular scale data.

Supplementary Material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.memlet.2025.100104.

Data availability

Data will be made available on request.