CO₂ Capture Characteristics of Hyperbranched Poly(alkylene imine): A Molecular Dynamics Simulation Approach

Abstract

This study explores the CO₂ capture characteristics of hyperbranched poly­(ethylenimine) (HB-PEI) and poly­(propyleneimine) (HB-PPI) through molecular dynamics simulations using density functional theory-calibrated force fields. Key features such as density, free volume, glass transition temperature, CO₂/H₂O distribution, and molecular diffusion are systematically investigated to elucidate structure–function relationships under dry and hydrated conditions. HB-PEI demonstrates a slightly higher density and lower free volume compared to HB-PPI yet shows superior CO₂ capture due to the high amine concentration. Glass transition analysis indicates a higher thermal mobility in HB-PEI, enhancing the CO₂ diffusivity. Pair correlation and coordination analyses confirm a stronger affinity of CO₂ with primary and secondary amines, particularly in hydrated environments where water competes with CO₂ for binding sites. Despite its more compact structure, HB-PEI outperformed HB-PPI in CO₂ and H₂O transport, as confirmed by higher diffusion coefficients across all hydration levels. These findings highlight a critical balance among polymer architecture, amine accessibility, and hydration in designing next-generation solid amine sorbents for efficient direct air capture applications.

1. Introduction

The drastic increase of the CO₂ concentration in the atmosphere due to human activities has raised temperatures by 1.1 °C over the last 150 years. Unless we reduce the emissions of CO₂ and other greenhouse gases significantly, it is expected that the average temperature will continue to rise, surpassing 2 °C by the end of the century, as reported by the Intergovernmental Panel on Climate Change. To prevent this undesirable process, it is necessary to reduce not only the emissions according to the roadmap of Net Zero Emissions by 2030 but also the overall concentration of CO₂ in the atmosphere, both of which can be accomplished using Carbon Capture and Storage (CCS) technologies. Carbon capture involves capturing CO₂ molecules via (1) direct air capture (DAC) and (2) separation of CO₂ from the exhaust gases emitted by industrial facilities (point source postcombustion capture). Recent studies have focused on developing advanced materials for DAC, addressing the challenge that CO₂ constitutes only about 0.04% of the atmosphere. These efforts concentrate on achieving selective CO₂ sorption in liquid-phase systems or solid sorbent materials. Although liquid amine absorption is mostly used in industrial flue gas separation, given the use of aqueous media with high heat capacity, it requires a lot of heat to recover the CO₂. , This means that conventional CCS processes are energy-intensive.

In this context, solid adsorption is being explored as a potentially economically advantaged choice. Xu et al. highlighted the advantage of the adsorption approach by noting its relatively low energy requirement and broad applicability across various temperature and pressure conditions. Similarly, numerous experimental studies with adsorbents, such as activated carbons , and metal oxides, , have demonstrated the applicability of adsorption for capturing CO₂ from gas mixtures. In addition, it is noted that multiple promising high-performing sorbents have also been studied through density functional theory (DFT) and molecular dynamics (MD) simulations. − Most studies describing DAC sorbent have focused on using solid-supported amine materials. ,,− Among these studies, solid amine sorbents based on porous oxide supports are the most common, in which CO₂ reacts with the amine functional group(s) and forms strong bonds, leading to substantial uptakes, even at low CO₂ partial pressures. Amine materials demonstrate superior uptakes associated with higher heats of sorption and better selectivities toward CO₂ compared to the physisorption-based sorbents, such as zeolites and metal–organic frameworks. Experimental studies have demonstrated that amine-functionalized sorbents can achieve CO₂/N₂ selectivity ratios exceeding 100 under dry conditions, driven by the formation of carbamate or bicarbonate species, which N₂ cannot form. , While our present simulation does not explicitly include N₂, the strong binding affinity and chemical specificity of amine–CO₂ interactions support the expected high selectivity. Several start-ups have sought to commercialize amine sorbent technologies, and Climeworks has developed a pioneering CO₂ capture facility in Iceland, named ORCA. Although the CO₂ uptake in a wide range of mesostructured materials has a high correlation with the surface area of the materials, it is noted that CO₂ uptake of amine sorbents is primarily determined by the concentration of accessible amine functional groups rather than simply the surface area, especially when the CO₂ concentration is less than 1.0 vol %.

Various strategies, including physical impregnation, covalent tethering, in situ polymerization, and their combinations, have been utilized to incorporate amine functionalities into porous support materials. Among these, the impregnation method involves depositing amine species onto the surface or within the pores of the solid support, enabling the incorporation of amine moieties without forming covalent bonds with the surfaces of materials. Low molecular weight poly­(ethylene imine) (PEI) has been employed in numerous studies as an amine-containing polymer to absorb CO₂ molecules because of its significant density of amine groups and good performance under temperature swing adsorption or vacuum swing adsorption (VSA) conditions. , However, PEI is susceptible to oxidative degradation at high temperatures, which diminishes its long-term efficiency. In this context, another polymer named poly­(propyleneimine) (PPI) has gained attention.

The chemical structure of PPI is similar to that of PEI, while its stability against oxidation is better than that of PEI, implying that PPI has a potentially longer operational lifetime than PEI. PPI possesses structural similarities to PEI, containing primary and secondary amines that interact with CO₂ to form an adsorbed species. It is expected that both PEI and PPI have similar reaction mechanisms through carbamate and bicarbonate formation reactions. However, their effectiveness ultimately depends on the variations in molecular architecture resulting from the differing carbon numbers in their respective backbones.

Experimental techniques such as Nuclear Magnetic Resonance (NMR) spectroscopy as well as X-ray and neutron scattering face challenges due to the complex polymer–support interactions in polymer-based CO₂ capture systems, limiting their ability to provide detailed and comprehensive understanding of CO₂ distribution and transport mechanisms. − These techniques have not fully elucidated the sorbed CO₂ distribution or transport mechanisms. MD simulation, however, can provide molecular-level insights into the structures and behaviors of the materials. Previous studies have shown that chain length and hydration conditions significantly affect CO₂ capture efficiency. Kim and co-workers, using MD simulations, observed that pair correlations between CO₂ and primary or secondary amines tend to decrease in hydrated environments, implying that carbamate formation may be less favored than under dry conditions. Regarding CO₂ capture, Sharma and colleagues utilized MD simulations to evaluate PEI nanostructures, concluding that free volume and entropy are key factors in predicting the effectiveness of PEI. Similarly, Shen et al. emphasized the role of reducing polymer chain length to enhance CO₂ capture efficiency in their MD simulation study. Additionally, it has been reported that both primary and secondary amines in hyperbranched poly­(ethylenimine) (HB-PEI) strongly associate with CO₂ molecules under dry conditions. However, under hydrated conditions, this interaction is reduced due to the association of amine groups with water molecules.

In this study, we investigate the distribution and transport of CO₂ molecules in both hyperbranched PEI and PPI systems using MD simulation to compare their CO₂ capture performance. For this purpose, we developed new force field parameters to achieve more accurate descriptions of molecular interactions among CO₂, water, and amines. The distribution and transport of CO₂ are characterized through various analyses such as pair correlation, coordination number, and mean-square displacement (MSD) analyses. Our study elucidates the microscopic behavior of these polymeric materials and thus gives insights into the design of new material systems with improved performance. While this work focuses on molecular-scale simulations of CO₂ transport and interaction, it is important to highlight the practical relevance of the studied materials, particularly HB-PEI, given its commercial availability and established industrial production.

2. Models and Simulation Methods

2.1. Molecular Models for HB-PEI and HB-PPI

We prepared HB-PEI and HB-PPI models, as shown in Figure a,b, respectively. It is noted that the ratio of primary/secondary/tertiary amines is 6:5:4 according to NMR results, and the molecular weights are 619.98 g/mol and 816.35 g/mol for HB-PEI and HB-PPI, respectively. The structures were geometrically optimized using the DFT method with B3LYP and 6-31G** through Jaguar. The atomic charges were assigned using Mulliken population analysis to individual atoms for calculating electrostatic interactions during the MD simulation.

1.

Molecular models for (a) hyperbranched poly­(ethylenimine) (HB-PEI) and (b) hyperbranched poly­(propyleneimine) (HB-PPI). Blue, orange, and green colors denote nitrogen atoms for primary amine, secondary amine, and tertiary amine, respectively, and gray and white colors denote carbon and hydrogen, respectively.

2.2. Force Fields for Polymer–CO₂–H₂O Interactions

In this study, the DREIDING force field and F3C force field were employed to describe HB-PEI, HB-PPI, and CO₂, and water. It is noted that the DREIDING force field has been employed in numerous simulation studies to investigate various materials in the literature, − whose results have been experimentally validated. − The DREIDING force field has the following form:

$$E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=E_{\mathrm{v}\mathrm{d}\mathrm{W}}+E_{\mathrm{Q}}+E_{\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{d}}+E_{\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}+E_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}+E_{\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}$$ 1

where E _total, E _vdW, E _Q, E _bond, E _angle, E _torsion, and E _inversion are the total, van der Waals, electrostatic, bond stretching, angle bending, torsion, and inversion energies, respectively. E _Q is calculated from atomic charges that are obtained from the Mulliken population analysis.

Particularly, in order to accurately describe interactions for amine–CO₂, amine–H₂O, and CO₂–H₂O pairs, first, we obtained the energy as a function of distance using the DFT method with B3LYP-D3 and 6-31G**, as shown in Figure a, and then determined the Lennard-Jones (LJ) potential parameters (D and r ₀) of eq for the off-diagonal van der Waals interactions (E _{off‑diagonal}):

$$E_{\mathrm{o}\mathrm{f}\mathrm{f}‐\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}}(r)=D(\frac{r_0^{12}}{r^{12}}-2\frac{r_0^6}{r^6})$$ 2

where D and r ₀ are the energy well depth and the distance at minimum energy, respectively. The LJ parameters are summarized in Tables S1 and S2 of the Supporting Information. The interaction (binding) energy between two species, A and B, was calculated using the following expression:

$${E_{\mathrm{B}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}}=E}_{\mathrm{A}+\mathrm{B}}-E_{\mathrm{A}}-E_{\mathrm{B}}$$ 3

where E _A+B is the total energy of the molecular pair optimized at the DFT level, and E _A and E _B are the energies of the isolated component. This definition allows for a direct comparison of DFT-derived interaction strengths and standard mixing rule predictions. Compared to the interaction energies calculated using the geometric-mean-based standard mixing rule, as shown in Figure S1, it is clearly demonstrated in Figure b,c that the interaction energy curves calculated using the newly determined LJ parameters are in good agreement with those from DFT calculations. The results clearly indicate that the overall mean-square error (MSE) calculated from using the newly optimized force field parameters was reduced by approximately 98% compared with the MSE calculated from using the force field parameters using the geometric-mean-based standard mixing rule. This significant reduction highlights the effectiveness of the alternative approach or method in achieving more accurate results. Therefore, the newly developed LJ parameters are used to perform the MD simulations in this study.

2.

(a) CO₂–amine pairs for DFT calculations. Binding energy curves calculated using newly determined Lennard-Jones parameters (b) for CO₂–amine and CO₂–C₂H₄ (c) for H₂O–amine pairs. The solid circles represent reference points obtained from DFT calculations. Blue, gray, red, and white colors denote nitrogen, carbon, oxygen, and hydrogen, respectively.

2.3. Bulk-Phase Model Preparation

The three-dimensional amorphous systems were constructed with the molecular models of HB-PEI or HB-PPI in a simulation box of 40 × 40 × 40 ų, with periodic boundary conditions in all directions, as shown in Figure a,b. The initial dimensions were adjusted during the equilibration process through NPT MD simulation. We also constructed various simulation systems using HB-PEI and HB-PPI with CO₂ and H₂O molecules. In detail, PEI-0-0 and PPI-0-0 have only HB-PEI and HB-PPI domains, respectively, without CO₂ and H₂O molecules, whose main purpose is to characterize the distribution of primary, secondary, and tertiary amines of HB-PEI and HB-PPI. In this study, first, we simulated 12, 36, and 72 CO₂ molecules in both HB-PEI and HB-PPI systems without water to investigate the distribution and transport of CO₂ molecules in the absence of moisture. Next, we simulated 72 CO₂ molecules with 36, 72, and 144 water molecules in the systems to investigate the effect of hydration on the amine–CO₂ interaction. Here, the choice of 72 CO₂ molecules corresponds approximately to a 1:1 ratio with the total number of reactive (primary and secondary) amine groups present in the system, facilitating direct comparison of amine–CO₂ interactions at saturation loading. Table summarizes the details of the simulated systems. For practical reference, we also calculated the gravimetric CO₂ content (mg of CO₂/g of adsorbent) for each system. For example, the PEI-72-0 system exhibits a CO₂ content of approximately 503 mg/g, assuming that all molecules are retained within the polymer domain. While this value offers a useful comparison point with experimental sorbents, it is important to note that the current simulations describe physical loading and do not incorporate chemical reaction mechanisms. Therefore, the reported CO₂ content reflects the total sorbate density rather than the reactive CO₂ uptake capacity.

3.

Simulated structures of amorphous bulk phases: (a) for hyperbranched poly­(ethylene imine) and (b) for hyperbranched poly­(propyleneimine). Blue, gray, and white colors denote nitrogen, carbon, and hydrogen, respectively.

1. Details of the Simulated Systems.

polymers (number of polymer)	systems	number of CO2	CO2 content (mg/g)	number of H2O	density (g/cm3)
HB-PEI (50)	PEI-0-0	0	0	0	0.843 ± 0.007
	PEI-12-0	12	17	0	0.851 ± 0.007
	PEI-36-0	36	51	0	0.869 ± 0.007
	PEI-72-0	72	102	0	0.894 ± 0.007
	PEI-72-36	72	102	36	0.919 ± 0.006
	PEI-72-72	72	102	72	0.939 ± 0.006
	PEI-72-144	72	102	144	0.978 ± 0.005
HB-PPI (38)	PPI-0-0	0	0	0	0.814 ± 0.007
	PPI-12-0	12	17	0	0.824 ± 0.007
	PPI-36-0	36	51	0	0.843 ± 0.006
	PPI-72-0	72	102	0	0.866 ± 0.006
	PPI-72-36	72	102	36	0.891 ± 0.005
	PPI-72-72	72	102	72	0.912 ± 0.005
	PPI-72-144	72	102	144	0.950 ± 0.005

2.4. Equilibrium MD Simulations of HB-PEI and HB-PPI

All MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator software. To achieve the equilibrium states of the simulated systems, we utilized a thermal annealing procedure outlined by Jang and Goddard, which accelerates the equilibration process by providing additional kinetic energy through the repetitive thermal process to overcome energy barriers from their trapped structures. Detailed procedure is described in the Supporting Information. No particular molecular or packed structures for HB-PEI and HB-PPI were assumed during the annealing procedure. Next, for 3-D systems, we performed NVT MD simulation for 200 ps and NPT MD simulation for 1 ns using the Nose–Hoover thermostat and barostat to complete the annealing procedure. Finally, we extended the NPT MD simulation for 60 ns at T = 303.15 K and P = 1 atm. On the other hand, for 2-D slab systems, we performed NVT MD simulation for 200 ps with the Nose–Hoover thermostat only to complete the annealing procedure. Finally, we extended the NVT MD simulation for 60 ns at T = 303.15 K. After completing the equilibrium MD simulation, we used the last 10 ns segment of the trajectory file for data analysis.

3. Results and Discussion

3.1. Force Field Development

Accurate prediction of intermolecular interactions is essential for capturing the behavior of HB-PEI and HB-PPI in the presence of CO₂ and H₂O. As described in Section , we refined the LJ parameters for the CO₂–amine, H₂O–amine, and CO₂–H₂O pairs so that the resulting interaction energies would closely match those obtained via DFT (B3LYP-D3/6-31G**) as shown in Figure . Notably, the refined parameters more accurately capture the critical short- and medium-range interactions relevant to CO₂ capture.

It is noted that (i) among the binding energies of CO₂ with amines (−5.31 kcal/mol, −5.78 kcal/mol, and −4.75 kcal/mol, for primary, secondary, and tertiary amines, respectively), the CO₂-secondary amine pair has the strongest binding energy, while the CO₂-tertiary amine pair has the weakest binding energy, and (ii) the binding energies of the CO₂–amine pairs are weaker than those of the H₂O–amine pairs (−10.67 kcal/mol, −11.10 kcal/mol, and −11.50 kcal/mol for primary, secondary, and tertiary amines, respectively).

A key outcome of the DFT-based parametrization is the capability to reproduce the observations from experimental and computational studies, ,,, that amine–H₂O interactions are stronger than amine–CO₂ interactions, validating the suitability of the newly developed force field for investigating CO₂ capture in both HB-PEI and HB-PPI.

3.2. Analysis of HB-PEI and HB-PPI in the Bulk Phase

3.2.1. Density and Free Volume Analysis

The initial density of both bulk-hyperbranched poly­(ethylenimine) (HB-PEI) and hyperbranched poly­(propyleneimine) (HB-PPI) was set to ρ₀ = 1 g/cm³. After the relaxation of the systems through NPT simulations, where the volume is allowed to change, the overall density changed significantly for both systems. The HB-PEI system achieved ρ_PEI = 0.843 g/cm³, and the HB-PPI system achieved ρ_PPI = 0.814 g/cm³. Since we obtained lower density values for the HB-PPI system compared to HB-PEI, we expected that the HB-PPI system has more free volume in the condensed bulk phase.

The free volume characteristics of HB-PEI and HB-PPI systems were analyzed using Connolly volumes and surfaces with probe radii of 1.4 and 2.0 Å, corresponding to the sizes of water and CO₂ molecules, respectively. As shown in Table and Figure a,b, using the 1.4 Å probe, HB-PEI exhibited a free volume fraction of 19.11%, while HB-PPI showed a slightly higher value of 19.88%. With the 2.0 Å probe, which better reflects the steric scale relevant to CO₂ diffusion, the free volume was reduced for both polymers to 6.86% for HB-PEI and 7.25% for HB-PPI, highlighting the exclusion effect for larger molecules. Specific surface area (SSA) measurements followed a similar trend, with HB-PEI showing 0.274 Å^–1 (1.4 Å probe) and 0.117 Å^–1 (2.0 Å probe), while HB-PPI showed 0.286 Å^–1 and 0.108 Å^–1, respectively.

2. Free Volume and SSA.

		free volume		specific surface area (Å–1)
polymer systems		1.4 Åprobe	2.0 Åprobe	1.4 Å probe	2.0 Å probe
HB-PEI	PEI-0-0	19.1%	6.9%	0.27	0.12
HB-PPI	PPI-0-0	19.9%	7.2%	0.29	0.11

4.

Connolly surface representation generated from the MD trajectory, illustrating the accessible void regions within the hyperbranched polymer system using a probe with a radius of 1.4 Å (a) for HB-PEI system and (b) for HB-PPI system. The gray and blue colors denote the outer and inner surfaces of the free volume, respectively.

These results indicate that HB-PPI possesses a slightly more accessible volume and surface area across both probe sizes. However, considering that each primary or secondary amine can donate an H-bond (via N–H) and any amine (including tertiary) can accept H-bonds via the lone pair on nitrogenand that interior amine groups are less available for interactionsit is plausible that HB-PEI forms a denser hydrogen bonding network and correspondingly a more compact internal structure. This structural compactness, driven by higher amine group density, explains why HB-PEI has a slightly higher density and lower free volume, particularly notable at the CO₂-sized probe scale. Despite this, the greater amine concentration in HB-PEI may impart superior CO₂-capturing ability, necessitating broader characterization of other physicochemical properties to compare the two systems fairly.

We also computed the radial distribution function (RDF) between hydrogens attached to nitrogen atoms (N–H donors) and neighboring nitrogen atoms (potential H-bond acceptors), thereby capturing the probability of the N–H···N hydrogen bond formation. We present these results as ρ·g­(r) curves to quantify the local density of hydrogen bonding interactions. As shown in Figures S4 (Supporting Information), HB-PEI consistently exhibits higher ρ·g­(r) intensity over the investigated distance range, indicating a higher probability of forming N–H···N hydrogen bonds. Particularly, HB-PEI shows a strong peak of ρ·g­(r) values in the 3.5–5.5 Å range, suggesting more extensive second-shell hydrogen bonding and intermolecular network formation.

This finding confirms that the higher amine concentration in HB-PEI facilitates a greater hydrogen bonding density, contributing to its slightly higher packing density and lower accessible free volume. At first glance, the densely packed structure of HB-PEI may seem unfavorable for CO₂ capture. However, as discussed in later sections, the higher segmental mobility and dynamic free volume in HB-PEI compensate for this static packing effect, enabling HB-PEI to exhibit a superior CO₂ transport and capture performance relative to HB-PPI.

3.2.2. Glass Transition Temperature

Since the CO₂ capture may depend on the thermal behavior of HB-PEI and HB-PPI through the CO₂ permeability, we performed a series of MD simulations to investigate the volume change as a function of temperature from 473 to 173 K. First, we equilibrated the HB-PEI and HB-PPI systems at 473 K, then decreased the temperature by 25 K, and ran NPT MD simulation for 30 ns at each temperature until we reached 173 K.

As shown in Figure , the volume decreased as the temperature was lowered for both systems. However, the slope of each system is significantly changed at ∼250 K, which is typical glass transition behavior from a rubbery to a glassy state. From the intersections of two slopes from high- and low-temperature regimes, it is found that T _g is 230.7 K (−42.5 °C) for the HB-PEI system and 240.1 K (−33.1 °C) for the HB-PPI system. The glass transition reflects a shift in the thermal expansion coefficient and corresponds to a change in heat capacity as a quasi-second-order phase transition. It is noted that the reported T _g of HB-PEI is ranged from ∼211.2 K (−62 °C) to ∼220.2 K (−53 °C), depending on the molecular weight, indicating that the simulated T _g values are comparable to the experimentally measured values.

5.

Volume change as a function of temperature for HB-PEI and HB-PPI systems.

Considering that the DAC process typically operates near-ambient conditions, the fact that T _g values of HB-PEI and HB-PPI are well below room temperature implies that they remain in a rubbery state, where high segmental motion promotes CO₂ diffusivity. This thermal molecular mobility thus enhances the dynamic accessibility of amine sites, aiding CO₂ transport and capture. It should be stressed that the T _g of HB-PPI is found to be slightly higher than that of HB-PEI, a difference attributed to the longer alkylene moiety in the molecular backbone, which results in increased molecular entanglements among branches, thereby restricting the mobility of the polymer chains. Therefore, it is inferred that HB-PEI is more favorable in the thermal behavior for CO₂ capture compared to that of HB-PPI.

3.3. Distribution of CO₂ and H₂O Molecules

3.3.1. Pair Correlation

To quantitatively characterize the distribution of CO₂ and H₂O, we used the pair correlation function for amine–CO₂, amine–H₂O, and H₂O–CO₂. The pair correlation function, g _A‑B ( r ), represents the probability density of finding A and B atoms at a distance (r), averaged over the equilibrium trajectory as shown in eq :

$$g_{\mathrm{A}-\mathrm{B}}(r)=\frac{n_B}{4\pi r^2\mathrm{d}r}/\frac{N_{\mathrm{B}}}{V}$$ 4

where n _B refers to the number of B particles located at a distance r from particle A within a shell of thickness dr, N _B is the total number of B particles in the entire system, and V represents the volume of the entire system. Please note, however, that eq is the ratio between the number densities of B particles in the shell and in the entire system. Since this g _A–B(r) is a normalized unitless quantity that converges to the 1.0 value for a random distribution of B particles, we used ρ_B·g _A–B(r), where ρ_B is the number density of B particles (ρ_B = N _B/V) to quantitatively compare the pair correlation:

$${\rho }_{\mathrm{B}}·g_{\mathrm{A}-\mathrm{B}}(r)=\frac{n_{\mathrm{B}}}{4\pi r^2\mathrm{d}r}$$ 5

Then, ρ_B·g _A–B(r) will exclusively account for the number density of B particles at the distance r from the central A particle, so high and low number density can be directly interpreted to have strong and weak correlation, respectively.

First, we investigated ρ·g(r) for amine–CO₂ pairs in HB-PEI and HB-PPI systems in the absence of H₂O molecules. N1, N2, and N3 denote the primary, secondary, and tertiary amines, respectively. From Figure a–f, we found that HB-PEI and HB-PPI show similar features: (1) the correlations become stronger with increasing the number of CO₂ molecules and (2) the CO₂ molecules have slightly closer access to the primary amines compared to the secondary amine at 2 Å < r < 4 Å, while the tertiary amine has clear correlation at 4 Å < r < 6 Å, indicating that the pair correlations for the amine–CO₂ pairs depend on the steric hindrance of the amine groups.

6.

Pair correlation functions for amine–CO₂ pairs in HB-PEI and HB-PPI bulk phases in the absence of H₂O molecules: (a,b) for PEI-12-0 and PPI-12-0, respectively; (c,d) for PEI-36-0 and PPI-36-0, respectively; and (e,f) for PEI-72-0 and PPI-72-0, respectively.

Next, we investigated ρ·g(r) for amine–CO₂ pairs in HB-PEI and HB-PPI systems in the presence of H₂O molecules. Figure a–f demonstrates how the amine–CO₂ pair correlations are affected by the water content. Overall, HB-PEI and HB-PPI have similar features. In detail, however, comparing PEI-72-0 and PPI-72-0, we found that the N1–CO₂ pair correlation becomes weaker at 2 Å < r < 4 Å but holds the distinctive intensity as a peak. In contrast, the N2–CO₂ pair loses the distinctive correlation at 2 Å < r < 4 Å but gains it at 4 Å < r < 6 Å. These features indicate that N2 is more affected by the presence of H₂O molecules compared to N1. Indeed, Figure S2 shows that the N2–H₂O pair correlation is distinctly increased as a function of water content, though N1–H₂O and N3–H₂O pair correlations also increased, simply less so. This means that the secondary amines are more solvated by water than the primary and tertiary amines.

7.

Pair correlation functions for amine–CO₂ pairs in HB-PEI and HB-PPI bulk phases in the presence of H₂O molecules: (a,b) for PEI-72-36 and PPI-72-36, respectively; (c,d) for PEI-72-72 and PPI-72-72, respectively; and (e,f) for PEI-72-144 and PPI-72-144, respectively.

3.3.2. Coordination Number and Vicinity Analysis

For further quantitative analysis, we calculated the CO₂ coordination number (CN) of the amine groups by integrating the first CO₂ shell around the amine groups

$$\mathrm{C}\mathrm{N}=4\pi \rho {\int }_0^{r_{1\mathrm{s}\mathrm{t}-\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}}}{r}^{2}g(r)\mathrm{d}r$$ 6

To quantify the degree of proximity between CO₂ molecules and amine functional groups, we computed the coordination numbers (CN) for primary (N1), secondary (N2), and tertiary (N3) amines across a series of HB-PEI and HB-PPI systems with varying CO₂ and H₂O contents (Table ). These coordination numbers reflect the likelihood of site-specific interactions that are essential for capturing CO₂ molecules, depending on the hydration level and amine type. For HB-PEI under dry conditions, CO₂ CN around the primary amines increases slightly with the CO₂ loading (from 0.75 at 12 CO₂ to 0.77 at 72 CO₂). The secondary amines exhibit an opposite trend, slightly decreasing from 0.64 to 0.61, whereas tertiary amines show a non-negligible decrease in CN (0.10–0.06). These trends are consistent with prior understanding that the primary and secondary amines are responsible for direct chemisorption via hydrogen bonding or nucleophilic attack, while the tertiary amines interact indirectly via bicarbonate formation.

3. CO₂ Coordination Numbers for Amine Groups in HB-PEI and HB-PPI Systems.

polymers	systems	CNCO2 (N1)	CNCO2 (N2)	CNCO2 (N3)
HB-PEI	PEI-12-0	0.75	0.64	0.10
	PEI-36-0	0.75	0.62	0.06
	PEI-72-0	0.77	0.61	0.06
	PEI-72-36	0.71	0.57	0.06
	PEI-72-72	0.71	0.51	0.06
	PEI-72-144	0.67	0.26	0.05
HB-PPI	PPI-12-0	0.64	0.54	0.03
	PPI-36-0	0.70	0.44	0.02
	PPI-72-0	0.63	0.45	0.04
	PPI-72-36	0.61	0.35	0.03
	PPI-72-72	0.60	0.33	0.02
	PPI-72-144	0.52	0.24	0.03

In hydrated systems, however, nuanced behavior emerges. For the PEI-72-X series (with increasing H₂O), the CN values for primary amines first drops (from 0.77 in dry to 0.67 at 144 H₂O), suggesting a subtle competition between CO₂ and H₂O at primary amine sites. The decrease in CN at intermediate hydration levels reflects the partial displacement of CO₂ due to water-mediated solvation or preferential hydrogen bonding. Interestingly, the CN values for secondary amines also decrease sharply from 0.61 (dry) to 0.26 (at highest hydration), while the CN values for tertiary amines remain nearly constant (∼0.06–0.04), supporting the hypothesis that CO₂ displaced from more reactive amine types may become transiently associated with sterically accessible tertiary sites.

In contrast, the HB-PPI system consistently exhibits lower CNs for all amine types compared with the HB-PEI system. For instance, CN_CO2 (N1) fluctuates around 0.63 under dry conditions and decreases with added water (0.63–0.52), which implies that HB-PPI might have lower CO₂ capturing capability, although it has slightly higher free volume. A similar trend holds for secondary and tertiary amines in HB-PPI, where the hydration leads to decreases in CN, albeit to a lesser extent than in HB-PEI. The limited variation implies that the weaker amine–CO₂ interactions in the HB-PPI are less sensitive to hydration, possibly due to the more hydrophobic and sterically hindered polymer environment.

3.4. Diffusion of CO₂ and H₂O Molecules

The efficacy of CO₂ capture in the HB-PEI and HB-PPI systems is significantly dependent on the transport of CO₂ and H₂O molecules since poor transport can hinder the adsorption–desorption process and adversely affect the overall performance. To quantify the transport behavior, we analyzed the MSD of CO₂ and H₂O molecules over time using eq :

$$\mathrm{M}\mathrm{S}\mathrm{D}=\frac{1}{N}\sum _{i=1}^N{|\mathit{r}(t)-\mathit{r}(0)|}^2$$ 7

where r (t) and r (0) denote the position of particle i at time t and the beginning and N denotes the number of particles. Figure a–f presents MSD change as a function of time for PEI-72-0, PEI-72-36, PPI-72-0, and PPI-72-36. Other systems in Table have a similar behavior to MSD. Accordingly, the diffusion coefficient (D) is defined as shown in eq :

$$D=\frac{1}{6N}\underset{t\to \infty }{\lim }\frac{1}{t}\sum _{i=1}^N{|{\mathit{r}}_i(t)-{\mathit{r}}_i(0)|}^2=\frac{1}{6}\underset{t\to \infty }{\lim }\frac{1}{t}\mathrm{M}\mathrm{S}\mathrm{D}$$ 8

8.

MSD for CO₂ and H₂O in HB-PEI and HB-PPI systems: (a,d) for CO₂ in the absence of H₂O; (b,e) for CO₂ in the hydrated systems; and (c,f) for H₂O in the presence of CO₂ in the hydrated systems.

Using this approach, we calculated the diffusion coefficients of CO₂ and H₂O, as summarized in Table . First, it is found that the CO₂ diffusivity is decreased with increasing number of CO₂ molecules in the dry systems. This result indicates that the molecular mobility in the dry systems is reduced due to the interaction of CO₂ with amine groups, which seems consistent with the carbamate formation between the CO₂ and amine groups acting as a physical cross-link that further restricts the CO₂ mobility.

4. Diffusion Coefficients of CO₂ and H₂O Molecules.

		diffusion coefficient (D)
polymer	systems	CO2 (×10–5 cm2/s)	H2O (×10–5 cm2/s)
HB-PEI	PEI-12-0	0.241 ± 0.054
	PEI-36-0	0.212 ± 0.010
	PEI-72-0	0.179 ± 0.003
	PEI-72-36	0.074 ± 0.002	0.058 ± 0.014
	PEI-72-72	0.042 ± 0.001	0.044 ± 0.002
	PEI-72-144	0.013 ± 0.001	0.022 ± 0.001
HB-PPI	PPI-12-0	0.110 ± 0.069
	PPI-36-0	0.098 ± 0.009
	PPI-72-0	0.097 ± 0.002
	PPI-72-36	0.049 ± 0.002	0.057 ± 0.011
	PPI-72-72	0.017 ± 0.001	0.021 ± 0.002
	PPI-72-144	0.009 ± 0.001	0.010 ± 0.001

Similarly, within the water content range (up to ∼0.2 H₂O/amine group), the CO₂ diffusivity is also decreased with increasing the number of water molecules because the interaction of H₂O with the amine groups reduces the molecular mobility in the systems. Notably, HB-PEI consistently exhibits higher CO₂ diffusivity than HB-PPI. For example, in the most hydrated system (72 CO₂ + 144 H₂O), the CO₂ diffusivity is 0.013 × 10^–5 cm²/s in the HB-PEI system compared to 0.009 × 10^–5 cm²/s in the HB-PPI system. Here, please note that HB-PEI is more thermally mobile than HB-PPI, as demonstrated in the glass transition analysis. It seems that the higher CO₂ diffusivity of HB-PEI is consistent with the greater thermal mobility of HB-PEI.

In hydrated systems, the diffusion coefficients of H₂O are statistically comparable to those of CO₂ under identical hydration conditions. This observation is consistent with expectations, as the H₂O molecules can have hydrogen bonding interactions with amine groups, as the CO₂ molecules have an affinity to the amine groups. In the HB-PEI system, the water diffusivity ranges from 0.058 × 10^–5 cm²/s at lower hydration (PEI-72-36) to 0.022 × 10^–5 cm²/s at higher hydration (PEI-72-144). A similar trend is observed in HB-PPI, where water diffusion decreases from 0.057 × 10^–5 cm²/s (PPI-72-36) to 0.010 × 10^–5 cm²/s (PPI-72-144). The reduction in the water diffusivity with increased hydration is likely due to the development of a water network that restricts translational motion and water cluster formation around amine groups, which reduce the molecular mobility of HB-PEI and HB-PPI.

Considering that HB-PPI has a higher free volume than HB-PEI, the simulation results seem counterintuitive at first glance. However, as reflected in the glass transition temperatures, HB-PEI has a greater thermal mobility than HB-PPI at the same temperature. Therefore, it is inferred that when the molecular interactions are significant, the molecular diffusivity depends more on thermal motions than on the free volume in the polymer system.

In addition to the molecular mobility of CO₂ and H₂O, the segmental mobility of the polymer matrix itself plays a crucial role in determining the transport efficiency. To further elucidate the connection between polymer dynamics and CO₂ diffusivity, we analyzed the MSD of nitrogen atoms (N) within the polymer backbone, which reflects the intrinsic segmental mobility of the amine-rich polymer network. Since nitrogen atoms are covalently attached to amine groups and distributed throughout the hyperbranched structure, their MSD provides a direct measure of the polymer matrix dynamics that modulates the dynamic free volume. As shown in Figure S3, the MSD of nitrogen atoms in HB-PEI is consistently higher than in HB-PPI across the full simulation time window. This result confirms that HB-PEI exhibits enhanced segmental mobility, which supports the formation of transient free volume pockets that facilitate molecular diffusion. This behavior is consistent with HB-PEI’s lower glass transition temperature and enhanced flexibility, as discussed in Section .

In contrast, the more rigid HB-PPI network produces fewer transient openings, limiting the CO₂ mobility. This additional analysis further supports the conclusion that HB-PEI’s superior CO₂ transport is not solely due to amine density or free volume but is enabled by the dynamic rearrangement of its polymer matrix. The rational design of future DAC sorbents should, therefore, account not only for chemical binding strength and static structure but also for segmental mobility and dynamic free volume formation.

4. Conclusion

In this study, we investigated the distribution and transport of CO₂ and H₂O molecules in the hyperbranched poly­(ethylenimine) (HB-PEI) and poly­(propyleneimine) (HB-PPI) through refined force field MD simulations. From analysis for the density, free volume, and SSA in bulk-phase HB-PEI and HB-PPI systems, we found that the HB-PEI system exhibits slightly higher density, lower free volume, and SSA compared to the HB-PPI system. This result is attributed to the higher amine concentration in HB-PEI than HB-PPI in the bulk phase, which results in a more developed hydrogen bonding network and molecular packing in the HB-PEI system compared to the HB-PPI system. Both polymers show glass transitions upon cooling, with HB-PEI transitioning at 230.7 K and HB-PPI transitioning at 240.1 K. This behavior aligns with their hyperbranched nature and indicates that HB-PEI has slightly more thermal mobility, which enhances the CO₂ diffusivity. Pair correlation analyses reveal that primary and secondary amines in both polymers have a high affinity for CO₂, especially under dry conditions. With hydration, water molecules compete with CO₂ for aminesmore significantly in secondary aminesthereby weakening CO₂ coordination. HB-PEI maintains stronger amine–CO₂ correlations overall. MSD and diffusion coefficient analysis show that HB-PEI consistently allows higher CO₂ mobility than HB-PPI across hydration levels up to ∼0.2 H₂O/amine group. In hydrated systems, water diffusion is also more favorable in HB-PEI. The enhanced mobility in HB-PEI stems from its higher thermal mobility, despite its lower free volume, which may facilitate efficient adsorption–desorption cycles. Collectively, these findings demonstrate that HB-PEI outperforms HB-PPI in CO₂ capture effectiveness. We expect that as additional experimental data are accumulated on HB-PPI, more details will be characterized, though less effective than HB-PEI under typical DAC conditions. Moreover, our simulations reveal that CO₂ transport remains effective under hydrated conditions, suggesting resilience to moisturea key challenge in real-world DAC operation. This humidity-tolerant behavior may extend the operational lifetime of these materials. Although a comprehensive life cycle assessment is beyond the current scope, future studies will aim to evaluate the carbon and energy costs of polymer synthesis with the long-term CO₂ removal capacity. Our study emphasizes the interplay between polymer structure, hydration, and thermal behaviors in designing next-generation sorbents for DAC applications.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.5c03162.

• Complete set of DFT-calibrated Lennard-Jones parameters and partial charges; binding energy curves calculated using the DFT method; pair correlation functions for amine–H₂O pairs in HB-PEI and HB-PPI bulk phases; MSD analysis of nitrogen atoms in HB-PEI and HB-PPI; RDF of N–H···N for hydrogen bonding interactions; and molecular model construction and equilibration (PDF)

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J.C. and G.R.W.N contributed equally to this work as cofirst authors.

The authors declare no competing financial interest.