Voids, or empty spaces within materials, exist across all length scales, from the cosmic voids influencing galaxy formation to interstitial voids in crystalline solids. At the nanoscale, nanovoids play a crucial role in the properties of materials, impacting catalysis, energy storage, separation processes, and medical applications. Their presence can alter surface area, create pathways for molecular or charge transport, enable compartmentalization, and modify mechanical responses. Despite their importance, understanding the morphology-function relationship in materials containing nanovoids is challenging due to limitations in imaging techniques. Macroscopic or micron-sized voids can be characterized using optical, ultrasound, and magnetic resonance imaging, but resolving the intricate nanomorphology of nanovoids requires nanometer-scale resolution and sufficient penetration depth. While X-ray tomography offers some capabilities, its resolution remains limited. Transmission electron microscopy (TEM)-based tomography, however, provides the necessary nanometer-scale resolution to image the complex 3D structures of materials like polyamide (PA) membranes. These PA membranes are critical components in thin-film composites used in water reclamation, molecular separation, and organic solvent nanofiltration. Previous studies have established quantitative relationships between synthesis conditions, morphology, mechanical properties, and separation performance of PA membranes, but primarily focused on simpler membranes with spatially separated voids. The more complex, commercially relevant membranes with interconnected networks of crumples and clustered nanovoids remain largely unexplored, particularly regarding their formation mechanisms and the impact on membrane properties. This study addresses this gap by integrating experimental techniques and simulations to elucidate the morphology-function relationships of PA membranes with interconnected nanovoids, utilizing graph theory to comprehensively capture the networked structure.

The formation of polyamide (PA) membranes via interfacial polymerization (IP) is a well-studied process, but the precise mechanisms driving the formation of the characteristic crumpled morphology and the associated nanovoids remain debated. Several hypotheses exist, including interfacial boiling, differential diffusion rates of monomers, local temperature increases, and nanobubbling of carbon dioxide. Some studies suggest an initial flat membrane acts as a self-limiting barrier, while others propose an incipient porous membrane that allows further monomer diffusion and reaction. The resulting nanovoids have been qualitatively observed in cross-sectional TEM and SEM images, and their influence on water permeability has been noted. However, a comprehensive, quantitative understanding of the 3D void structure and its relation to membrane properties remains lacking. Existing morphometry techniques are often limited to simpler structures, and a robust method for characterizing the complex interconnected networks of voids found in advanced PA membranes is needed. The application of graph theory to describe materials networks, while growing, has largely focused on 2D images; extending these methods to 3D electron tomography data of soft materials is a significant challenge. Previous work has successfully employed graph theory to analyze various materials, from organic molecules and nanoparticles to biological structures, but adapting these techniques to capture the complexity of interconnected nanovoids in PA membranes requires innovative approaches.

This research employed a multifaceted approach combining experimental techniques and computational simulations. Polyamide (PA) membranes were synthesized via interfacial polymerization of trimesoyl chloride (TMC) and m-phenylenediamine (MPD) with varying monomer concentrations (PA1, PA2, and PA3). Low-dose electron tomography using TEM, with a dose rate of 4–7 e⁻ Å⁻² s⁻¹, was employed to obtain high-resolution 3D images of the membrane nanostructures, mitigating beam damage. A total of 61 TEM images were collected for each sample, covering a tilt range from -60° to +60° in 2° increments. These images were then aligned and processed using IMOD and OpenMBIR software to generate 3D tomograms. Custom morphometry analysis was developed to quantify the local thickness of the membrane, excluding the void spaces. This method involves inscribing the largest possible sphere within the membrane at each voxel, and the diameter of this sphere is defined as the local thickness. Three-dimensional void reconstruction was performed by segmenting the PA material from the void spaces in the tomograms. This approach allowed for the quantification of various void parameters, including void volume, void ratio, and void volume fraction. Two types of voids were distinguished: open voids connected to the membrane surface and closed voids completely enclosed by the PA material. Coarse-grained molecular dynamics (CGMD) simulations, using the model developed by Muscatello et al., were employed to simulate the interfacial polymerization process and investigate the mechanism of nanovoid formation. The simulations used coarse-grained representations of the monomers, allowing for efficient simulation of the reaction-diffusion processes. The CGMD simulations were run for 600 ns with a time step of 3.0 fs. Graph theory (GT) analysis was applied to the skeletonized 3D tomograms to characterize the overall structural rigidity of the membrane. Skeletonization reduced the 3D tomographic data to a 1D network of nodes and branches. Various GT parameters, including graph density (ρ) and global efficiency (E), were calculated to quantify the structural connectivity and load transfer efficiency of the network. Atomic force microscopy (AFM) was used to measure the apparent modulus of the membranes, providing a measure of their nanomechanical properties. The Spiegler-Kedem model was modified to incorporate the 3D void morphology parameters, including local thickness variations and void surface areas, to predict methanol permeance. X-ray photoelectron spectroscopy (XPS) was used to determine the degree of crosslinking (DOC) of the membranes. Atomistic simulations were also performed to study the influence of different TMC:MPD ratios on the modulus of PA membranes under compression loading. The mechanical properties were calculated using NEMD simulations under a constant strain rate in a NσɛT ensemble.

The study revealed several key findings: 1. **Nanovoid Formation Mechanism:** Three-dimensional electron tomography and CGMD simulations demonstrated that interconnected nanovoids in PA membranes form through a coalescence and growth mechanism of oligomers. Thicker regions of the membrane arise from the merging of thinner regions, providing strong experimental support for the simulation results. The local thickness of the membranes showed a multimodal distribution with three maxima (t1, t2, t3), consistent with the coalescence of oligomers. 2. **Void Morphology and Methanol Permeance:** The volume-filled void reconstruction method allowed for the quantification of void morphology in 3D. Increased TMC monomer concentration led to a decrease in both void volume and void ratio. This detailed 3D void information, incorporating open and closed void surface areas and local thickness variations, proved crucial for accurate fitting of the experimentally measured methanol permeance using a modified Spiegler-Kedem model. The model incorporating void structure significantly improved the accuracy of permeance predictions compared to using a simple average membrane thickness from AFM. 3. **Graph Theory and Nanomechanical Properties:** The skeletonization of the tomograms, coupled with GT analysis, provided a quantitative measure of the membrane's structural rigidity. Graph density (ρ) and global efficiency (E) correlated strongly with the apparent modulus measured using AFM. Membranes with denser and more efficient networks (higher ρ and E) exhibited higher apparent moduli. This finding suggests that the interconnectedness of the crumple network, influenced by the TMC:MPD ratio, plays a significant role in determining the membrane's mechanical strength. The trend in apparent modulus observed experimentally was also confirmed by atomistic IP models subjected to compression loading in MD simulations. 4. **Tunable Morphology and Properties:** The TMC monomer concentration acted as a key control parameter, influencing various aspects of membrane morphology, including void volume, crumple density, local thickness, and degree of crosslinking. The results demonstrate the capability of tuning the final membrane properties through control of the monomer concentrations during synthesis.

This research successfully addressed the challenge of understanding the complex relationship between nanovoid morphology and the functional properties of PA membranes. The integrated approach of experimental 3D imaging, computational simulations, and graph theory analysis provided a comprehensive understanding of the nanovoid formation mechanism and its impact on both separation performance and nanomechanical properties. The finding that void morphology significantly influences methanol permeance has important implications for membrane design and optimization in various applications. The strong correlation between graph theory parameters and apparent modulus provides new insights into the structure-property relationships in these materials, offering opportunities to design membranes with tailored mechanical properties. The tunability of membrane properties by varying monomer concentrations provides a practical approach to control membrane performance. Future studies could explore other types of polymer membranes, investigating the generality of the oligomer coalescence mechanism and the applicability of the combined approach to other nanomaterials. Exploring the impact of different solvent systems and reaction conditions on nanovoid formation would further enhance our understanding of this complex process. The development of more sophisticated graph theory metrics could potentially capture additional aspects of the network structure and its relationship to membrane properties.

This study provides a comprehensive understanding of the formation and functional relevance of nanovoids in polyamide membranes. The integration of advanced imaging techniques, computational modeling, and graph theory analysis revealed a novel oligomer coalescence-and-growth mechanism for nanovoid formation. Void morphology was shown to be a critical determinant of both permeance and mechanical properties. The results highlight the power of a multi-scale approach to understand complex nanomaterials, opening avenues for designing membranes with improved performance. Future work could focus on expanding this approach to other membrane systems and exploring the potential of targeted void engineering to control membrane properties.

While the study provides substantial insights, some limitations exist. The CGMD simulations, while efficient, simplified some aspects of the complex interfacial polymerization process, such as the exact role of solvents and the influence of local temperature variations. The Spiegler-Kedem model, while effective in capturing the permeance trend, may not fully capture all aspects of mass transport in these complex materials. The AFM measurements provided apparent moduli, which are influenced by both the intrinsic material properties and the complex nanostructure; further work is needed to fully decouple these effects. The analysis focused on methanol permeance, and future studies should evaluate other solvents to assess the generality of the findings.