Double-Gyroid Network Morphologies Formed by Asymmetric AB₁B₂ Triblock Amphiphiles over Wide Volume Fraction Range

Abstract

Molecular dynamics simulations are used to investigate the phase behavior of asymmetric AB₁B₂-type miktoarm triblock amphiphiles, composed of a sugar-based acyclic headgroup (A) and two hydrocarbon tails (B₁ and B₂). AB₁B₂ amphiphiles with significantly shorter B₂ tails (f _B1 /F _B2 ≫ 1, where f _i is the volume fraction) form lamellar (LAM) and perforated lamellae (PL) structures, whereas those with nearly equal tail lengths (f _B2 ≈ F _B1 ) assemble into hexagonally packed cylinders (CYL). Amphiphiles with a B₁/B₂ length ratio near 2:1 (2f _B2 ≈ F _B1 ) stabilize double gyroid (DG) networks, where the headgroups form the interconnected channels and the tails constitute the matrix, displaying feature sizes from 1.7 to 3.3 nm across a broad volume fraction range with $0.22\le f_{\mathrm{A}}\le 0.40$ . For potential applications in membrane separation at infinite dilution, these networks significantly hinder the diffusion of polar molecules, while nonpolar molecules diffuse relatively unimpeded. Diffusion selectivities near 3 are found for 1-butanol versus water and n-hexane versus methanol. Self-consistent field theory (SCFT) calculations corroborate the presence of DG networks at intermediate compositions for AB₁B₂ miktoarm triblock polymers, although no specific B₁/B₂ ratio is predicted to significantly broaden the network phase window. This study highlights the role of asymmetry in the molecular design of amphiphilic block oligomers, which enables the stabilization of network morphologies with ultrasmall feature sizes over a wide composition range.

Introduction

The exploration of the self-assembly of block polymers and amphiphilic oligomers has garnered significant attention due to their broad applications as thermoplastic elastomers, , templates for mesoporous ceramics, , and microelectronics patterning materials. − Among these, smaller amphiphiles, also known as high-χ block oligomers, are particularly promising as they can form morphologies with feature sizes smaller than 5 nm, − extending the typical application range of block copolymers, which generally exhibit periodic patterns with length scales of 10–50 nm. − This capability opens up new possibilities for nanofiltration, nanopatterning, and other nanoscale applications. −

Self-assembly of block polymers and amphiphilic oligomers can lead to diverse morphologies, including lamellae (LAM), hexagonally packed cylinders (CYL), body-centered cubic micellar (BCC), and network (NET) structures. − Among these, NET morphologies, characterized by interpenetrating and percolating nanodomains, , have attracted interest for a range of applications, including nanoporous membranes, − ion transport media, − and therapeutic delivery vehicles. , Despite their potential, stabilizing NET structures remains challenging due to packing frustration, which generally restricts their existence to narrow composition windows. , In double gyroid (DG) networks, the curvature near the gyroid nodes typically exhibits a saddle-like shape with negative Gaussian curvature. In contrast, for the gyroid struts, the curvature is approximately cylindrical, characterized by a Gaussian curvature that approaches zero, though it may exhibit slight local variations due to the geometric complexity of the network. , Stabilizing DG structures requires molecular architectures that can accommodate these curvature requirements, posing a significant design challenge. Various strategies, such as blending molecules that exhibit different morphologies, designing novel architectures such as ABC triblock polymers, coil–brush/bottlebrush polymers, − and glycolipids, − have been explored to broaden the stability window of NET morphologies.

Among the diverse applications of NET structures, those with smaller feature sizes are particularly valuable for nanofiltration and nanopatterning, driven by the demand for miniaturization. However, the large chain length (N) of polymers limits the attainable pore diameter or unit cell dimension in NET structures. Approaches to achieving smaller feature sizes have focused on selecting blocks with higher effective interaction (χ) parameters (i.e., unlike interactions that are less favorable than like interactions), enabling stabilization of NET morphologies in low-N block polymer systems. Amphiphiles, characterized by a high χ parameter between hydrophilic and hydrophobic segments, along with their typically low molecular weight (low N), are well-suited to stabilize NET structures with smaller feature sizes. This stabilization potential has been demonstrated in various amphiphilic systems, including bolaamphiphiles and glycolipids, underscoring the promise of high-χ, low-N materials. − Our previous studies, using both simulations and experiments, examined amphiphilic block oligomers with sugar-based acyclic headgroups (A) and hydrocarbon tails (B), revealing mesophases with remarkably small domain sizes. − Notably, Shen et al. showed that blending LAM-forming AB amphiphiles (with one tail) and CYL-forming AB₂ amphiphiles (with two equal-length tails) can stabilize DG morphologies with feature sizes as small as 2.2 nm. However, NET morphologies in neat systems of hydroxylated block oligomers have yet to be reported.

In this study, we employ molecular dynamics (MD) simulations to investigate the self-assembly of solvent-free amphiphilic block oligomers into various morphologies. An asymmetric molecular design transforms AB₂ amphiphiles (with two B tails of equal length) into AB₁B₂ amphiphiles, composed of a hydrophilic oligool headgroup (H _n , with n = 4, 6, or 8, containing n – 1 −CHOH- repeat units and one −CH₂OH terminal unit) and two hydrophobic alkyl tails (T _l and T _s ), each consisting of −CH₂– repeat units, a −CH₃ terminal unit, and a CH connector to the polar head (Figure ). Here, the number of carbons in the longer tail is denoted by l, while that in the shorter tail is represented by s. We extensively explore molecular designs for H₄T­(T _l )­(T _s ) amphiphiles with $s/l$ tail length ratios ranging from 0.1 to 0.9, and l + s ranging from 6 to 21. To validate that the resulting design principles extend to larger oligomers, we also investigate H₆T­(T₁₈)­(T₉) and H₈T­(T₁₈)­(T₉) amphiphiles. Furthermore, we investigate the potential of the DG network structures for membrane-based separation by analyzing the tracer diffusion of water, methanol, 1-butanol, n-hexane, and ions (Li⁺, F^–, Na⁺, Cl^–). In addition, we perform self-consistent field theory (SCFT) calculations on AB₁B₂ miktoarm triblock polymers to compare the microphase separation in flexible block polymers with the strongly enthalpy-driven self-assembly in stiff, high-χ block oligomers.

1.

Morphologies of asymmetric H₄T­(T _l )­(T _s ) amphiphiles (where l and s indicate number of carbon atoms in the long and short tail, respectively) predicted from MD simulation at T _SIM = 430 K for l + s ≤ 10 and T _SIM = 445 K for $l+s>10$ . Colors represent distinct morphologies: orange for double gyroid (DG), blue for hexagonally packed cylindrical (CYL), cyan for perforated lamellar (PL), purple for lamellar (LAM), and gray for disordered (DIS) structures. Dashed lines in blue, cyan, and gray mark the boundaries of DG and other phases, with the orange dotted line indicating a 2:1 tail length ratio.

Methods

Molecular Models and Simulation Details

The interactions of the H _n T­(T _l )­(T _s ) amphiphiles are modeled using the transferable potentials for phase equilibria united-atom (TraPPE–UA) force field. − For diffusion studies of small molecules and ions, water is represented by the SPC/E model, 1-butanol, methanol, and n-hexane by the corresponding TraPPE–UA models, and ions (Li⁺, F^–, Na⁺, Cl^–) by single-site models with parameters from the AMBER03 force field. Additional details on the molecular models and force field parameters are provided in the Supporting Information and Tables S1 and S2. MD simulations are conducted in the isobaric–isothermal (NpT) ensemble using GROMACS 2021.3. , System sizes range from 1000 to 2500 molecules. Temperature and pressure are controlled by the Nosé-Hoover thermostat , with a relaxation constant ${\tau }_T=0.4$ ps and the Parrinello–Rahman barostat with a relaxation constant ${\tau }_p=2$ ps, respectively. Electrostatic interactions are computed via the particle-mesh Ewald method, while the p-LINCS algorithm constrains O–H bond lengths, enabling a 2 fs time step. Initial disordered configurations are generated through short Monte Carlo (MC) simulations in the canonical ensemble at T _SIM = 3000 K. Since the coupled-decoupled configurational-bias MC moves applied to the united-atom model do not maintain tacticity, each amphiphile includes a random mixture of stereoisomers. Systems are equilibrated at p = 1 atm and temperatures ranging from T _SIM = 430 K for n = 4 and l + s ≤ 10 to T _SIM = 520 K for H₈T­(T₁₈)­(T₉) using MD trajectories of at least 1 μs. Additional trajectories of 200 ns for LAM, PL, and CYL phases and 400 ns for DG phases are performed for structural analysis. Simulation snapshots showing surface meshes and ball-and-stick representations are visualized using Ovito and VMD. −

System Size Tuning and Workflow

Initial simulations for all AB₁B₂ molecules are conducted in orthorhombic cells with 1000 or 2000 molecules, allowing independent fluctuations in the x-, y-, and z-dimensions to mitigate incommensurability effects. , This setup is suitable for morphologies with one- or two-dimensional periodicity, such as LAM and CYL structures, where domain spacing can adjust through rotations and lateral expansions or contractions perpendicular to the lamellar planes or cylinder axes. For three-dimensionally periodic structures (e.g., BCC and NET), however, the 1000-molecule AB₁B₂ systems exhibit locally segregated but globally disordered networks due to persistent incommensurability unless the simulation box contains integer multiples of the unit cell. To achieve ordered NET structures, it is essential to know the lattice parameter (a) and the number of molecules per unit cell (N _UC) to prevent distorted, metastable configurations. − The unit cell dimension a is estimated by calculating the structure factor S(q) for disordered NET-like structures and using the broad peak position: $a=2\pi m/q^{*}$ , where m corresponds to the first observable reflection spacing ratio for the morphology (e.g., $m=\sqrt{6}$ for DG). N _UC is then estimated based on the average molecular volume from initial simulations. For each AB₁B₂ system forming disordered NET structures, simulations are restarted with 8N _UC molecules in a cubic box of length L _box = 2a, allowing for ordered NET phases such as double gyroid (DG), double diamond (DD), and single gyroid (SG). To facilitate ordering, we employ a workflow for 3D NET simulations that includes a guiding field, created by Gaussian interaction sites with different strengths for H and T beads, distributed within the minority and majority domains of candidate NET structures. , After removing the guiding field, the stability of each candidate NET morphology is assessed over a 1 μs trajectory. Values of N _UC and molecular volume for each system are provided in Table S3.

Tracer Diffusion Analysis

The diffusion analysis is based on the mean squared displacement (MSD), defined as $⟨\mathrm{\Delta }{r}^2(t)⟩=⟨(r(t)-r(0){)}^2⟩$ , where r(t) is the position of the tracer particle at time t, and angle brackets denote the statistical average. Following classical diffusion theory and assuming Brownian motion, the MSD is expected to increase linearly with time, enabling the calculation of the diffusion coefficient D from the slope: MSD(t) = 6Dt. To simulate infinite-dilution conditions, two molecules or ion pairs are introduced into each simulation box containing a preassembled DG structure, and molecular dynamics simulations of 1 μs are conducted to compute the MSD. , Diffusion coefficients for the guest molecules are obtained by linearly fitting the MSD curves over the range from 10³ ps to 10⁵ ps, and uncertainties are estimated as the bootstrap standard error.

SCFT Calculations

SCFT is known for accurately predicting block polymer phase behavior and providing theoretical insights into the formation of complex network phases. − In this paper, SCFT calculations are performed to map the phase behavior of AB₁B₂ star polymers across a range of compositions, with a focus on the effects of volume fraction f _A and asymmetry ratio $f_{\mathrm{B}1}/f_{\mathrm{B}2}$ at a fixed $\chi N$ ; notably, when $f_{\mathrm{B}1}/f_{\mathrm{B}2}=0$ , the system reduces to a linear AB diblock copolymer. All SCFT calculations are conducted using the open-source Polymer Self-Consistent Field (PSCF) software developed by Morse and co-workers. Polymers are modeled as Gaussian chains with equal statistical segment lengths for A and B segments unless otherwise specified. The calculations are performed with an integration step size ds = 0.01, a convergence tolerance of 10^–6, and spatial grid resolutions of 64n (where n = 1,2, or 3 for 1D, 2D, and 3D calculations, respectively). Initial guesses for ordered phases, including LAM, CYL, DG, PL, and O ⁷⁰ phases, are generated using symmetry-adapted basis functions to facilitate convergence. PSCF then iteratively updates these initial configurations to obtain self-consistent solutions, with Anderson mixing applied to enhance convergence stability, enabling precise free energy comparisons to identify equilibrium phases.

Results and Discussion

Phase Behavior of AB₁B₂ Amphiphiles

The morphologies of solvent-free asymmetric two-tail amphiphilic oligomers H₄T­(T _l )­(T _s ) are investigated using MD simulations at T _SIM = 430 K for l + s ≤ 10 and T _SIM = 445 K for $l+s>10$ . A sequential simulation workflow is employed to determine the number of molecules in the simulation box, ensuring that the number of molecules matches the unit cell size of the corresponding morphology. In this workflow, a guiding field is applied to direct the initial configuration toward various network phases, followed by long MD simulations to assess their stability (see details in the Methods section).

The phase behavior of the asymmetric two-tail amphiphilic oligomers investigated in this work is summarized in Figure . We explore a broad design space, ranging from the smallest H₄T­(T₄)­(T₂) to the largest H₄T­(T₁₄)­(T₇), where the volume fractions of the polar block are f _A = 0.43 and f _A = 0.19, respectively, and we observe a variety of stable morphologies.

Starting with nearly symmetric AB₁B₂ amphiphilic oligomers with tails of nearly equal length, we observe that H₄T­(T₈)­(T₇) and H₄T­(T₁₀)­(T₉) self-assemble into stable CYL structures. For H₄T­(T₈)­(T₇), the static structure factor numerically calculated from the MD simulation exhibits characteristic peaks consistent with the CYL morphology, specifically at ratios of $\sqrt{1}$ : $\sqrt{3}$ : $\sqrt{4}$ (Figure a; details of the structure factor calculation are provided in the Supporting Information). To further illustrate the morphology, we present dividing surfaces between the polar (A block) and nonpolar (B blocks) domains, along with molecular configurations in ball-and-stick representations (Figure b). Amphiphiles with slightly smaller $s/l$ ratio, such as H₄T­(T₈)­(T₆), H₄T­(T₁₀)­(T₈), and H₄T­(T₁₀)­(T₇), are still able to maintain CYL structures (Figure a), though defects appear, primarily in the form of cylinder bridging and undulated bending of the cylinders (Figure S1).

2.

(a) Static structure factors, S(q), of H₄T­(T₈)­(T _s ) amphiphiles for s values from 1 to 7, with labels indicating the corresponding d-spacings. (b) Molecular configuration snapshots in ball-and-stick representations (white: hydroxyl hydrogen; red: oxygen; gray: CH _x unit), alongside dividing surfaces between polar and nonpolar regions, for the equilibrium morphologies of H₄T­(T₈)­(T₇), H₄T­(T₈)­(T₄), H₄T­(T₈)­(T₂), and H₄T­(T₈)­(T₁).

For AB₁B₂ amphiphiles with a B₂/B₁ ratio close to 1:2, stable DG structures are observed. Figure a shows the structure factors of DG-forming H₄T­(T₈)­(T₅), H₄T­(T₈)­(T₄), and H₄T­(T₈)­(T₃), with characteristic peak position ratios at $\sqrt{6}$ , $\sqrt{8}$ , $\sqrt{16}$ , $\sqrt{20}$ , $\sqrt{22}$ , and $\sqrt{24}$ , consistent with the Ia3̅d (Q ²³⁰) space group symmetry. Simulation snapshots of H₄T­(T₈)­(T₄) show eight unit cells in the simulation box (Figure b). As illustrated in Figure , the observed range of stable DG phases around the 1:2 ratio (orange dotted line) highlights the generality of this asymmetric molecular design. We find DG stability over a relatively wide volume fraction window, from the smallest H₄T­(T₄)­(T₃) and H₄T­(T₅)­(T₂) (f _A = 0.40) to the largest H₄T­(T₁₂)­(T₆) (f _A = 0.22). To further test the ability of the 1:2 asymmetric design to stabilize DG structures, we probe two larger molecules, H₆T­(T₁₈)­(T₉) and H₈T­(T₁₈)­(T₉), using MD simulations at correspondingly higher T _SIM (Figures S2 and S3). The observed DG structures indicate that even for AB₁B₂ molecules with larger headgroups and longer tails, the 1:2 design principle remains effective. The molecular weights and numbers of dihedral angles for the DG-forming AB₁B₂ amphiphiles (M _w = 234 and 634 g/mol for H₄T­(T₄)­(T₃) and H₈T­(T₁₈)­(T₉), respectively) span the molecular weight range of other sugar-based block oligomers that have been reported to form ordered phases. ,,,

Additionally, we examine the phase behaviors of the DG-forming H₄T­(T₈)­(T₅) and H₄T­(T₈)­(T₄) amphiphiles at T _SIM = 460 K. Interestingly, H₄T­(T₈)­(T₄), with a B₂/B₁ ratio of 1:2, forms a disordered structure at this elevated temperature, while H₄T­(T₈)­(T₅), slightly deviating from the 1:2 ratio, self-assembles into a stable CYL structure (Figure S4).

For molecules with a smaller s, such as H₄T­(T₈)­(T₂) and H₄T­(T₁₀)­(T₂), self-assembly into perforated lamellae (PL) occurs (Figure ). Simulation snapshots of H₄T­(T₈)­(T₂) further confirm this structure (Figure b). The static structure factor, S(q), for H₄T­(T₈)­(T₂) shows a second peak close to the primary peak, indicating the average lateral domain spacing between neighboring perforations (Figure a). Both H₄T­(T₈)­(T₂) and H₄T­(T₁₀)­(T₂) form PL structures with perforations arranged in a nearly hexagonal order; however, this order is not fully stable, as some disorder in the perforation arrangement remains. Therefore, we refer to these structures as PL rather than hexagonally perforated layers (HPL). Figure S5 shows simulation snapshots for the H₄T­(T₁₀)­(T₂) systems, where dividing surfaces of the primary layers reveal perforations with approximately hexagonal ordering, albeit with visible disorder. For molecules with an even smaller s, such as H₄T­(T₈)­(T₁) and H₄T­(T₁₀)­(T₁), LAM structures are observed, with S(q) peak positions matching the characteristic ratios of 1:2:3 (Figure ).

The domain spacing for AB₁B₂ amphiphiles can be determined from the structure factor using $d=2\pi /q^{*}$ , where $q^{*}$ is the position of the primary peak. For CYL, DG, and PL/LAM structures, this corresponds to d ₁₀, d ₂₁₁, and d ₁₀, respectively, representing domain spacings in specific directions. The domain spacings for the ordered morphologies of the amphiphiles studied here range from 1.7 to 3.3 nm (Table S3). For the DG structures formed by the H₄T­(T _l )­(T _s ) amphiphiles, the domain spacing increases approximately linearly with the total number of carbon atoms in the alkyl tails, i.e., l + s + 1 (Figure a). For l + s + 1 ≤ 14, d ₂₁₁ increases rapidly with total tail length at a rate of 0.09 nm per carbon atom. For $l+s+1>14$ , however, the rate the d ₂₁₁ increase slows to 0.03 nm per carbon atom as the tails become more flexible. Comparing among different morphologies found for the H₄T­(T _l )­(T _s ) amphiphiles, the domain spacing decreases for all morphologies as f _B decreases (i.e., decreasing total length of the tail segments for constant number of headgroup segments) and, for similar f _B, the domain spacing increases in the order LAM < PL < DG < CYL (Figure b).

3.

Domain spacing and morphology observed for H₄T­(T _l )­(T _s ) amphiphiles: (a) Equilibrium domain spacing d ₂₁₁ for DG morphologies as a function of the number of carbon atoms in the alkyl tails (l + s + 1) at T _SIM = 430 K and T _SIM = 445 K; (b) morphology (symbol type) and domain spacing (symbol color) as functions of the volume fraction of the polar A block and the volume fraction ratio $f_{{\mathrm{B}}_2}/f_{{\mathrm{B}}_1}$ .

Level Surface Analysis

To further assess the quality of the DG structures formed by the H₄T­(T _l )­(T _s ) amphiphiles, a level surface analysis is applied. − The DG morphology can be approximated by the following level surface:

$$\sin \left(2\pi \frac{x}{L}\right)\cos \left(2\pi \frac{y}{L}\right)+\sin \left(2\pi \frac{y}{L}\right)\cos \left(2\pi \frac{z}{L}\right)+\sin \left(2\pi \frac{z}{L}\right)\cos \left(2\pi \frac{x}{L}\right)=t$$ 1

where x, y, and z are spatial coordinates, L is the unit cell size, and t is the level surface constant, which determines the volume ratio between the gyroid channels and the matrix. To analyze the spatial distribution of molecules within DG morphologies, the coordinates of all oxygen atoms in the simulation box are substituted into the level surface equation, and the distribution of t is calculated as a quality indicator of DG structures. Values of $t\ge 1$ correspond to the right-handed channel, values of $t\le -1$ correspond to the left-handed channel, and values in between $-1<t<1$ correspond to the matrix between these two channels. Figure a shows the distribution of t for three representative DG morphologiesH₄T­(T₄)­(T₃), H₄T­(T₈)­(T₄), and H₄T­(T₁₂)­(T₆)with distinct positive and negative peaks, confirming the self-assembly of molecules into DG structures. For all DG structures investigated in this work, the distributions of level surface constant t show that less than 40% of t values for the oxygen atoms fall within the range $t\in [-1,1]$ (Figure S6), suggesting that the majority of the hydroxyl groups in DG-forming amphiphiles are positioned predominantly within the two gyroid channels.

4.

(a) DG level surface analysis of oxygen atom coordinates in H₄T­(T₄)­(T₃), H₄T­(T₈)­(T₄), and H₄T­(T₁₂)­(T₆) amphiphile systems. The gray dashed lines at $t=\pm 1$ indicate the approximate boundaries between the two channels and the matrix, with the region $t\in [-1,1]$ corresponding to the matrix, and regions outside this range corresponding to the channels. (b) The normalized density of oxygen atom coordinates obtained from three-dimensional kernel density estimation for H₄T­(T₈)­(T₄). Heatmaps showing (c) analytical results along the (111), (110), and (211) directions, (d) data for H₄T­(T₄)­(T₃), and (e) data for H₄T­(T₁₂)­(T₆). Color bars indicate t values.

Moreover, a 3D kernel density estimation is applied to compute the relative density of oxygen atom coordinates for H₄T­(T₈)­(T₄) (Figure b). This analysis reveals that regions of high oxygen atom density are concentrated within the gyroid channels. Figure S7 displays a scatter plot in which points representing densities above the 95th percentile of the overall distribution are highlighted in red, corresponding to normalized density values ranging from 0.87 to 1. Analyzing the distribution of t values for oxygen atoms at different densities further reveals that most oxygen atom coordinates are clustered in the upper-right region, indicating that high-density points are located within the skeletal graph regions of the gyroid channels (Figure S8). Within these DG structures, the interpenetrating channels are visualized using a volumetric density map along three characteristic directions, based on interpolated oxygen atom coordinates and continuous mapping with the level surface as a scalar field (Figure d,e). The resemblance of these volumetric maps to the analytical heatmap slices from the level surface equation confirms that the internal DG structure is well-preserved (Figure c). Notably, structures, such as the double diamond, single gyroid, and plumber’s nightmare (also known as the double primitive), generated through system-size tuning and guiding fields, rapidly become disordered within 50 ns after the guiding field is removed, eliminating them as candidates for stable NET structures (Figure S9).

To characterize the spatially segregated and bicontinuous morphologies in greater detail, a density-based spatial clustering of applications with noise (DBSCAN) approach is employed to analyze the spatial distribution of oxygen atoms within the DG phases. Figure displays the clustering outcome for the final frame of the simulation trajectory for the H₄T­(T₄)­(T₃) system, where periodic boundary conditions extend atom coordinates to neighboring cells, ensuring the inclusion of atoms near the box boundaries in the DBSCAN analysis. The clustering analysis reveals two well-defined clusters (Figure a). To validate that these clusters represent distinct gyroid channels, the level surface analysis is conducted (Figure b), which illustrates that the oxygen atoms in each cluster are consistently excluded from the vicinity of t = 0, demarcating the matrix that separates the two gyroid channels, thereby verifying that these clusters correspond to spatially distinct, left- and right-handed channels. The inset in Figure b shows that each cluster contains a nearly equivalent number of points, indicating an approximately balanced occupancy of H₄T­(T₄)­(T₃) molecules within each channel in the DG phase. Additionally, the DBSCAN analysis for the H₄T­(T₈)­(T₃) system (with a larger number of carbon atoms in the longer alkyl tail) identifies a small number of minor clusters and noise points (Figure S10), which account for fewer than 1% of all data points, confirming the high structural quality of the DG phase formed by the H₄T­(T _l )­(T _s ) amphiphiles.

5.

(a) DBSCAN clustering analysis of oxygen atom coordinates in the equilibrium morphology of H₄T­(T₄)­(T₃). (b) Level surface analysis of oxygen atom coordinates for the two clusters identified through clustering. The inset shows the number of data points in each cluster.

Amphiphile Packing

To investigate the molecular packing mechanisms of the amphiphiles in greater detail, the angular distribution between tail vectors in the simulation box is analyzed (Figure ). Given that the H₄T­(T _l )­(T _s ) amphiphiles possess the same compact tetraol headgroup (H₄), with a volume fraction under 40% for ordered phases, we hypothesize that the relative orientation between vectors along the tails significantly influence the local interfacial curvature between the polar and nonpolar domains. Since each H₄T­(T _l )­(T _s ) molecule features two tails of differing lengths and the morphology varies with the shorter tail length, as demonstrated in Figure , our analysis focuses on the angles formed between the longer tails of adjacent molecules (Figure ). For the angular distribution analysis, we selected H₄T­(T₈)­(T _s ) molecules that assemble into distinct morphologies, specifically H₄T­(T₈)­(T₁), H₄T­(T₈)­(T₂), H₄T­(T₈)­(T₃), and H₄T­(T₈)­(T₇). The tail vector of each molecule is defined as the vector extending from the midpoint of the C–C bond that joins the A and B blocks to the terminal methyl group of the longer tail (Figure e). The angle θ between the longer tails of any two neighboring molecules is defined by the angle between their respective vectors. A coordination sphere cutoff corresponding to each molecule’s d-spacing is applied, with specific cutoff distances for H₄T­(T₈)­(T₁), H₄T­(T₈)­(T₂), H₄T­(T₈)­(T₃), and H₄T­(T₈)­(T₇) being 2.56, 2.19, 2.03, and 1.93 nm, respectively. Neighboring molecules are defined based on the distance between the starting points of their tail vectors.

6.

Zoomed-in simulation snapshots of (a) LAM, (b) PL, (c) DG, and (d) CYL structures formed by H₄T­(T₈)­(T₁), H₄T­(T₈)­(T₂), H₄T­(T₈)­(T₃), and H₄T­(T₈)­(T₇), respectively. Amphiphiles are shown as non-overlapping spheres (white: hydroxyl hydrogen; red: oxygen; gray: CH _x unit). The dividing surfaces are shown in the colors used for the ADF. (e) Angular distributions of end-to-end vectors for the longer tail groups between neighboring molecules in H₄T­(T₈)­(T₁), H₄T­(T₈)­(T₂), H₄T­(T₈)­(T₃), and H₄T­(T₈)­(T₇) systems. The inset illustrates the definitions of the vectors and angle.

The calculated angular distribution functions (ADF) reveal distinct packing preferences associated with different morphologies (Figure e). For the LAM-forming H₄T­(T₈)­(T₁) amphiphile, there is a strong preference for both parallel and antiparallel packing (peaks at 0^◦ and 180^◦, respectively), consistent with the characteristics of the lamellar morphology (Figure a). The ADF for the LAM morphology exhibits a global minimum near 90^◦, indicating that the short tail of the H₄T­(T₈)­(T₁) molecules does not support perpendicular packing. As the coordination sphere cutoff matches the domain spacing d ₁₀, the angular distribution primarily reflects a balance between parallel and antiparallel packing within layers, with minor contributions from molecules in adjacent layers favoring antiparallel packing. This results in an asymmetric ADF curve, displaying a slightly higher probability density for antiparallel packing compared to parallel packing, as shown in Figure e. For the PL-forming H₄T­(T₈)­(T₂) amphiphile, parallel and antiparallel packing is also preferred due to the layered structure. Compared to the LAM morphology, the perforated nature of the PL morphology (Figures b and S5) leads to a greatly diminished preference for parallel and/or antiparallel packing and an enhanced tendency for perpendicular packing within the coordination sphere for molecules situated at the high-curvature perforations.

For the CYL-forming H₄T­(T₈)­(T₇) amphiphiles, the headgroups pack into cylindrical arrangements, with the tails radiating uniformly in all directions, as illustrated in Figure d. This results in a more evenly distributed ADF with minor local maxima at the parallel and antiparallel orientations (Figure e) that is surprisingly similar to the ADF for the PL morphology. In contrast, the ADF for the DG structure formed by the H₄T­(T₈)­(T₃) amphiphile exhibits a nearly uniform distribution with a slight, approximately linear decrease from antiparallel to parallel angles, lacking a local minimum near 90^◦, distinguishing it from the ADFs of other structures. The ADF for the DG morphology contains a noticeably higher proportion of perpendicular packing near 90^◦ as compared to the minima observed for the other morphologies. This increased perpendicular orientation arises from the approximately 1:2 ratio of the short to long tails in the H₄T­(T₈)­(T₃) amphiphiles, allowing the short tail to more effectively support perpendicular packing between neighboring amphiphiles. From a morphological perspective, due to the complex, interpenetrating network and the absence of planar regions in the DG structure, amphiphiles are less likely to adopt parallel or antiparallel packing within the coordination sphere defined by the d ₂₁₁ spacing.

For comparison, the ADF values at $\cos (\theta )=-1$ , 0, and 1 obtained for the H₄T­(T₈)­(T _s ) amphiphiles (with s = 1, 2, 3, and 7) are presented in Figure S11. For angles between long tails corresponding to parallel $(\cos (\theta )=1)$ and antiparallel $(\cos (\theta )=-1)$ orientations, the relative intensities follow the same order: the LAM structure formed by the H₄T­(T₈)­(T₁) amphiphiles shows the highest probability density, followed by the PL structure found for H₄T­(T₈)­(T₂), then the CYL structure found for H₄T­(T₈)­(T₇), and the DG structure formed by the H₄T­(T₈)­(T₃) amphiphiles exhibits the lowest probability density. This ordering indicates that the DG morphology least favors antiparallel packing and actually disfavors parallel orientations. For perpendicular orientation $(\cos (\theta )=0)$ , the DG morphology shows the highest probability density, followed by CYL and PL, while the LAM morphology displays a far lower values, suggesting a slightly stronger preference for perpendicular packing in the DG structure compared to the other three ordered morphologies. Additionally, the ADF calculated using a tighter coordination sphere cutoff of 0.7 nm is shown in Figure S12. This cutoff is selected based on previous studies of mixtures of H₄T­(T₈)­(T₂) and H₄T­(T₈). Using this smaller cutoff to focus on pairs of amphiphiles that are in closest proximity (e.g., mostly belonging to the same leaflet for LAM), the LAM structure formed by H₄T­(T₈)­(T₁) shows a strong preference for parallel alignment. Under this condition, the ADF profiles of the PL structure formed by H₄T­(T₈)­(T₂) and the CYL structure formed by H₄T­(T₈)­(T₇) become more similar to that of the DG structure formed by H₄T­(T₈)­(T₃), with the DG ADF lying between those of PL and LAM.

Tracer Diffusion for Membrane-Based Separation Applications

Membrane-based separation is a highly promising application area for NETs due to their potential to facilitate selective transport across nanoscale domains. ,− Here, we investigate the potential of the DG structures formed by H₄T­(T _l )­(T _s ) amphiphiles for membrane-based separation applications. We hypothesize that molecules with different affinities for the DG channel, matrix, and interfacial regions can be separated due to different transport resistances of these different regions. 1-butanol has received attention as biofuel with superior fuel qualities compared to ethanol, but its separation from water, the fermentation solvent, is particularly challenging due to low concentration. , Another industrially important separation is the azeotrope-forming methanol/n-hexane mixture that can be aided by a pervaporation process. Ion selective membranes are desired for efficient lithium extraction from salt-lake brines.

Here, we calculate the tracer diffusion (see Methods) for water, 1-butanol, methanol, n-hexane, and LiF and NaCl ion pairs in the H₄T­(T₄)­(T₃), H₄T­(T₈)­(T₄), and H₄T­(T₁₂)­(T₆) systems (Figure ; numerical values are reported in Table S4). These structures, with d-spacings ranging from 1.67 to 2.39 nm and volume fractions for the polar block from 0.40 to 0.22, provide a comprehensive evaluation of how the characteristic length scales of self-assembled structures affect tracer diffusion. The mean-square displacements of the guest molecules (but not ions, see below) calculated from the simulation trajectories exhibit linear time dependence, with a log–log slope close to 1, indicating normal diffusion behavior (Figures S13–S15). For all four guest molecules, the diffusion coefficients in the H₄T­(T₄)­(T₃) system are significantly lower than in the two systems with the longer alkyl tails (Figure a). The slightly lower temperature (430 versus 445 K) likely contributes to this trend, but it also appears that the H₄T­(T₄)­(T₃) system forms a more rigid structure with higher transport resistances. Interestingly, the H₄T­(T₄)­(T₃) system is moderately selective for water over 1-butanol with a selectivity factor, S _WB (where $S_{ij}=D_i/D_j$ the ratio of the diffusion coefficients of species i and j), of 1.5, whereas the H₄T­(T₈)­(T₄) and H₄T­(T₁₂)­(T₆) systems allow for faster 1-butanol than water diffusion with S _BW = 2.5 and 2.4, respectively. Due to its smaller size, methanol diffuses faster than 1-butanol for all three systems. With regard to the methanol/n-hexane separation, the diffusion selectivity for n-hexane increases with increasing volume fraction of the nonpolar part of the amphiphiles (S _HM = 1.2, 1.5, and 3.1 for H₄T­(T₄)­(T₃), H₄T­(T₈)­(T₄), and H₄T­(T₁₂)­(T₆) systems, respectively).

7.

(a) Tracer diffusion coefficients for water, 1-butanol, methanol, and hexane in the DG structures formed by H₄T­(T₄)­(T₃) at T _SIM = 430 K, H₄T­(T₈)­(T₄) at T _SIM = 445 K, and H₄T­(T₁₂)­(T₆) at T = 445 K. (b) Level surface analysis of the spatial distribution of tracer center-of-mass positions over 1 μs simulation trajectories in the three DG structures. Proportion indicates the duration that molecules/ions remain in three distinct regions, i.e., $|t|<1$ , $t\ge 1$ , and $t\le -1$ , corresponding to the gyroid matrix, right-handed gyroid channel, and left-handed gyroid channel, respectively.

For the ions Li⁺, F^–, Na⁺, and Cl^–, the slopes obtained from the log–log plots of MSD versus time deviate significantly from unity, indicating that their diffusion does not conform to normal (Brownian) behavior for the system sizes and simulation length used here (Figures S16–S18). Analyzing the MSD via a generalized diffusion equation yields subdiffusive behavior (Figure S19), presumably because ion transport involves infrequent node-to-node hopping events. The extent of the ion preference for nodes with their higher concentration of hydroxyl groups will be sensitive to the ion–hydroxyl group interactions, and fixed-charge ion models with slightly reduced charges (e.g., the Madrid ion models) may yield faster ion dynamics.

To further elucidate the transport mechanism, a level surface analysis is employed to assess the relative positions of molecules and ions within the DG structures over the entire 1 μs simulation trajectory. At each time point, the coordinates for every interaction site of a given guest molecule or ion are used to compute a corresponding level surface constant, t, and these are used to obtain the distribution of t values (Figure S20). According to the DG level surface equation, the simulation box can be partitioned into three distinct regions (the gyroid matrix for $|t|<1$ , the right-handed gyroid channel for t > 1, and the left-handed gyroid channel for $t<-1$ ), and the fractional time that the tracers spend in these three regions can be calculated (Figures b and S19b). For the polar molecules, the fraction of time spent in the channels increases with increasing length of the tails of the H₄T­(T _l )­(T _s ) amphiphiles, and this trend is most pronounced for water. Concomitantly, the preference for n-hexane to reside in the matrix increases. We surmise that increasing the alkyl tail length leads to more pronounced local segregation of the polar and nonpolar blocks. The three polar guest molecules exhibit relatively symmetric distributions (Figure S20) and roughly equal probabilities for residing in the left- and right-handed channels (Figure b). This indicates that these molecules with sufficient frequency move between left- and right-handed channels. In contrast, a symmetric distribution is not achieved for the ions and, in particular, Li⁺ and F^– ions predominantly remain in either the left- or right-handed channel. As the d-spacing increases, the channels become more widely separated and the free energy barrier for transport through the matrix increases, thereby making it increasingly difficult for ions to escape a given channel. It remains an open question whether larger system sizes (i.e., larger number of ions leading to a more equal distribution over the two channels) or longer simulation trajectories would qualitatively change the anomalous diffusion behavior observed here for the ions.

Phase Behavior of Asymmetric AB₁B₂ Block Polymers from SCFT Calculations

A comparison of the SCFT phase diagram for asymmetric AB₁B₂ block polymers with that for the AB₁B₂ amphiphilic oligomers is presented in Figure . The SCFT calculations begin with conformationally symmetric linear AB diblock copolymers near the network phase boundary, transitioning to an AB₁B₂ star triblock by introducing a second B block while maintaining a constant total B segment length (B₁ + B₂ = B). Figure a depicts the phase diagram at $\chi N=30$ , with an additional phase diagram at $\chi N=15$ shown in Figure S21, where the selected $\chi N$ match the range obtained by fitting the order–disorder transition boundary from the MD simulations of the two-tail amphiphilic oligomers from our previous work. A notable observation is that the DG network phase region shifts toward higher f _A values as the B₂/B₁ ratio increases. When the B₂/B₁ ratio is increased, the volume fractions of B₂ and B₁ segments become more comparable, resulting in a more compact arrangement of the B segments. This configuration alters the spontaneous interfacial curvature, even though the total volume fraction of the B segments remains constant. Specifically, an increased length of the B₂ segment lowers the spontaneous curvature, requiring a higher volume fraction of the A segment to balance it. Consequently, this shifts the network phase window toward higher f _A values. However, no specific B₂/B₁ ratio is found to significantly expand the network phase region; only a slight change in the gyroid phase width is observed around a B₂/B₁ ratio of 0.3, though this variation is minor. Interestingly, at $\chi N=15$ , a small region of stable O ⁷⁰ morphology appears between the DG and LAM phases in the phase diagram (Figure S21).

8.

Phase diagrams as a function of the volume fraction of the polar A block and the volume fraction ratio $f_{{\mathrm{B}}_2}/f_{{\mathrm{B}}_1}$ for (a) AB₁B₂ star polymers from SCFT calculations at $\chi N=30$ and (b) AB₁B₂ amphiphilic oligomers from MD simulations.

Another notable distinction between the systems of polymers and amphiphilic oligomers is the absence of a stable HPL/PL phase. In the case of the block polymer, the absence of the HPL phase can be expected when the A blocks form the perforated layers. For AB₂ star polymer melts, SCFT predicts a narrow composition window in which the HPL morphology is stable at high segregation when B is the minority block. However, when A becomes the minority block, the entropic penalty associated with stretching the B blocks to fill the majority-component layers renders the HPL phase unfavorable in the AB₂ miktoarm architecture, resulting in reduced stability compared to linear diblock polymer melts. For AB₂ star copolymers with a majority A block, which resembles a large headgroup, a stable HPL phase can form with B blocks packed into the perforated layers, which is less relevant to the phase diagram of amphiphilic oligomers shown in Figure b. Consequently, we do not expect an HPL phase in AB₁B₂ systems where A is the minority block. This suggests that the presence of the PL phase in the block oligomer system is enthalpically driven, as the entropic chain-stretching penalties that inhibit the PL phase in the block polymer system are much less significant for the stiffer oligomer chains. To explore the effect of conformational asymmetry, a phase diagram at $\chi N=15$ with $b_{\mathrm{B}}<b_{\mathrm{A}}$ is calculated (Figure S22). While the f _A windows broaden slightly, changes in conformational symmetry do not significantly affect the influence of tail-length ratiosthe f _A range remains nearly constant at lower $f_{{\mathrm{B}}_2}/f_{{\mathrm{B}}_1}$ values and expands as $f_{{\mathrm{B}}_2}/f_{{\mathrm{B}}_1}$ increases.

Conclusion

MD simulations are employed to investigate the phase behavior of AB₁B₂ miktoarm triblock amphiphilic oligomers, which consist of polar, sugar-based (A) and nonpolar, hydrocarbon (B) segments with asymmetric tails. With different $f_{{\mathrm{B}}_2}/f_{{\mathrm{B}}_1}$ ratios, DG, LAM, PL, and CYL morphologies are observed. DG structures can be stabilized across a wide range of A block volume fractions (f _A) from 0.22 to 0.40 when $f_{{\mathrm{B}}_1}\approx 2f_{{\mathrm{B}}_2}$ . Domain spacings are found to range from 1.67 nm for H₄T­(T₄)­(T₃) to 3.26 nm for H₈T­(T₁₈)­(T₉). DG structures formed by amphiphiles with longer alkyl tails (i.e., higher f _B) provide well-defined pathways for transport of less polar and nonpolar molecules which in the bulk would be miscible with alkanes, whereas transport of strongly polar molecules is hindered. This allows for faster transport of larger nonpolar molecules, and diffusion selectivities near 3 are observed for 1-butanol versus water and n-hexane versus methanol.

SCFT calculations also predict the DG window for AB₁B₂ star triblock polymers, although no significant broadening of the DG window occurs at specific B₂/B₁ ratios. This work demonstrates the value of asymmetric molecular design in stabilizing DG structures, but also underscores the differences in network phase window formation between stiff block oligomers and flexible block polymers.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacsau.5c00466.

• Additional force field details; details on structure factor calculation; and supplementary tables and figures (PDF)

CRediT: Daoyuan Li data curation, formal analysis, investigation, methodology, visualization, writing - original draft; Zhengyuan Shen investigation, methodology; Pengyu Chen data curation, formal analysis, investigation, methodology, writing - original draft; Mahesh K. Mahanthappa formal analysis, writing - review & editing; Kevin D. Dorfman formal analysis, supervision, writing - review & editing; Timothy P. Lodge conceptualization, formal analysis, supervision, writing - review & editing; J. Ilja Siepmann conceptualization, formal analysis, supervision, writing - review & editing.

The authors declare no competing financial interest.