Introduction

Block copolymers (BCPs) are composed of chemically distinct polymer blocks covalently linked along a single chain and are well known for their capacity to self-assemble into a range of well-ordered nanostructures1,2. The self-assembly behavior originates from the balance between segmental interactions and chain entropy, resulting in equilibrium morphologies such as lamellae, cylinders, spheres, and bicontinuous networks3,4. This morphological diversity enables their widespread application in advanced technologies, including directed self-assembly, membrane fabrication, and energy storage devices5.

The thermodynamics of BCP self-assembly is typically characterized by the Flory–Huggins interaction parameter (\(\chi\))6,7, which quantifies the enthalpic incompatibility between unlike polymer segments. For symmetric AB diblock copolymers, the equilibrium morphology is mainly determined by the product \(\chi N\) and the block volume fraction (\(f\))8. Phase diagrams constructed from these parameters serve as the basis for predicting phase behavior9,10. Classical approaches, notably self-consistent field theory (SCFT), are widely used to predict phase boundaries within the mean-field approximation11,12. However, SCFT does not account for fluctuation effects or local compositional heterogeneities, thus limiting its accuracy for finite-chain lengths and near the order–disorder transition (ODT)5. Moreover, the \(\chi\) is often treated as an effective parameter that incorporates ill-defined entropic and non-ideal contributions, which can result in inconsistencies between theoretical predictions and experimental observations13,14. In particular, for systems where experimental access to microscopic dynamics is limited, computational modeling serves as a valuable complementary approach to address these limitations.

Recent advances in polymer synthesis, such as anionic polymerization and reversible addition–fragmentation chain transfer (RAFT) polymerization, have enabled the preparation of BCPs with low dispersity and well-defined molecular weight and composition15,16,17. These developments have broadened the accessible parameter space for block copolymer phase behavior, permitting systematic variation of architectural and chemical features beyond classical scaling relations (e.g., D ∼ N2/3)18,19. In parallel, advanced characterization techniques such as small-angle X-ray scattering (SAXS)20, transmission electron microscopy (TEM)21, and neutron reflectometry (NR)22 have provided detailed insights into morphology, including measurements of domain spacing (D), interfacial width, and ODT temperatures (TODT)23. Despite these advances, experimental investigations remain limited in resolving transient morphological states, defect-mediated transitions, and local structural dynamics, particularly under nonequilibrium conditions24. As a result, molecular simulations serve as a complementary approach to investigate self-assembly mechanisms at the molecular scale25.

Computational approaches are widely employed to connect molecular-level interactions with mesoscale structure formation26. Commonly used techniques include self-consistent field theory (SCFT), dissipative particle dynamics (DPD)27, and lattice-based Monte Carlo simulations28. SCFT provides efficient predictions of equilibrium morphologies but does not capture thermal fluctuations or chain entanglement effects29. DPD captures the kinetics of self-assembly but typically employs soft-core interactions that do not fully represent excluded volume effects and topological constraints30. Lattice-based Monte Carlo methods are suitable for examining thermodynamic transitions31; however, spatial discretization limits their capacity to represent continuous chain conformations. These considerations have motivated the development of physically informed reduced-order models that preserve chain-level resolution and allow simulations at extended length and time scales.

Coarse-grained molecular dynamics (CG-MD) simulations implement chain connectivity through harmonic bond potentials and, when required, angular (and dihedral) potentials to modulate chain stiffness32. Nonbonded interactions are modeled using truncated Lennard-Jones (LJ) potentials, parameterized by interfacial interaction strength (\({\varepsilon }_{\left(A,B\right)}\)) and cutoff distance (\({r}_{c,\left(i,j\right)}\))33. Chain length (N) is systematically varied to examine its influence on morphology9. Despite their physical realism, coarse-grained (CG) models generally lack a direct correspondence to experimentally accessible thermodynamic parameters. Establishing quantitative relationships between CG interaction parameters and the Flory–Huggins paramter (χ) remains a major challenge for the predictive design of block copolymer systems34. Recent computational studies have emphasized the importance of tuning \({\varepsilon }_{(A,B)}\) and \({r}_{c,(B,B)}\) to modulate interfacial interaction strength and control morphological evolution35. Morphological transitions among spherical, cylindrical, and lamellar phases can be induced by changes in molecular architecture (e.g., from linear to Y-branched topologies), even at fixed segmental volume fractions, suggesting that conformational entropy penalties can dominate interfacial curvature under strong interfacial interaction strength conditions36. To address these limitations, establishing quantitative relationships between LJ interaction parameters and effective segregation strength is necessary for predictive modeling and the rational design of block copolymer materials.

In this study, CG-MD simulations are employed to investigate the self-assembly and morphological characteristics of symmetric AB diblock copolymers, with a particular focus on lamellae structures. The effects of chain length (N = 20, 50, 80), the cutoff distance for B–B interactions (\({r}_{c,(B,B)}\) = 1.5, 2.0, 2.5), and the A–B interfacial interaction strength (\({\varepsilon }_{(A,B)}\) = 0.5–0.9) are examined (Fig. 1). The structural organization has been characterized by one-dimensional descriptors such as density profiles and the lamellar spacing (D). Additional analyses involve three-dimensional metrics, such as intermaterial dividing surface (IMDS) mapping, density projections, structure factor S(q*), and spatial autocorrelation functions. Quantitative relationships between microscopic interaction potentials and segregation strength are established by correlating these structural metrics with interaction parameters. This approach provides a computational framework for understanding and controlling morphology in block copolymer systems and offers a basis for extending these strategies to more complex architectures, including multiblock copolymers, branched topologies, and polymer–nanoparticle hybrids.

Fig. 1: Schematic illustration of the CG models of BCPs and key molecular parameters in the simulations.
Fig. 1: Schematic illustration of the CG models of BCPs and key molecular parameters in the simulations.The alternative text for this image may have been generated using AI.
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a The simulation begins from a random initial configuration and equilibrates into a phase-separated morphology. b Three key parameters are systematically varied to investigate their impact on microphase separation.

Results

Morphological evolution and one-dimensional segment density profiles

The effects of interfacial interaction strength (\({\varepsilon }_{\left(A,B\right)}\)) and B-block interaction range (\({r}_{c,\left(B,B\right)}\)) on the phase-separated morphologies of AB diblock copolymers were examined. Representative morphologies for copolymers with chain length N = 80 are shown in Fig. 2. In these simulations, the cutoff distances for A–A and A–B interactions are fixed at \({r}_{c,\left(A,A\right)}={r}_{c,\left(A,B\right)}=2.5\), while \({r}_{c,\left(B,B\right)}\) was varied from 1.5 to 2.5 and \({\varepsilon }_{\left(A,B\right)}\) from 0.5 to 0.9. As shown in Fig. 2, the resulting morphologies reflect a balance between interfacial tension and cohesive interactions. Low miscibility between the blocks favors phase separation, resulting in the formation of ordered lamellar structures37. When \({{\rm{\varepsilon }}}_{\left(A,B\right)}\) is increased, interfacial attraction limits the growth of domains and facilitates block intermixing38. A larger \({r}_{c,\left(B,B\right)}\) enables more stable domain formation and a higher degree of microphase separation. Previous simulations and field-theoretic analyses indicated a relationship between \({\varepsilon }_{(A,B)}\) and the Flory–Huggins parameter χ, which is often approximated as \(\chi \approx 0.39\,{\varepsilon }_{(A,B)}-1\)39. Accordingly, \({\varepsilon }_{(A,B)}\) serves as a practical proxy for thermodynamic incompatibility. Recent Monte Carlo studies further suggest that the effective χ parameter scales inversely with temperature as \(\chi =\alpha /T\), where α depends on interaction range and cohesive strength40. Additionally, first-order perturbation theories such as the Weeks–Chandler–Andersen approach, can be applied to directly calculate effective incompatibility parameters from CG potentials, showing consistent trends across different chain lengths.

Fig. 2: Microphase-separated morphologies of symmetric AB diblock copolymers under varying interfacial interaction strength and segmental cohesion.
Fig. 2: Microphase-separated morphologies of symmetric AB diblock copolymers under varying interfacial interaction strength and segmental cohesion.The alternative text for this image may have been generated using AI.
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Each row corresponds to a different homotypic cutoff radius for B segments: a \({r}_{c,\left(B,B\right)}=1.5\), b \({r}_{c,\left(B,B\right)}=2.0\), and c \({r}_{c,\left(B,B\right)}\) = 2.5. In all cases, the cutoff distances for A–A and A–B interactions are fixed at \({r}_{c,\left(A,A\right)}={r}_{c,\left(A,B\right)}\) = 2.5. Each snapshot shows the steady-state configuration of the system, with A-type beads colored green and B-type beads colored orange.

Comparisons between longer (N = 80) and shorter chains (N = 20, 50; Figs. S1 and S2 in the Supplementary Information) indicate clear improvements in domain regularity and continuity with increasing chain length. Specifically, the improved regularity observed at N = 80 arises from reduced entropic fluctuations associated with increased chain stretching, a mechanism that has been previously documented in CG simulations41. This suppression of fluctuations stabilizes the lamellar interfaces, resulting in more ordered and continuous domains at higher interaction ranges (\({r}_{c,(B,B)}\)). These observations align well with experimental studies that attribute improved morphological order in longer chains to a more effective balance between enthalpic and entropic contributions42.

Morphologies of symmetric AB diblock copolymers with chain length N = 80 under various inter-block interaction strengths \({\varepsilon }_{\left(A,B\right)}=0.5-0.9\). Each row corresponds to a different homotypic cutoff radius for B segments: (a) \({r}_{c,\left(B,B\right)}=1.5\), (b) \({r}_{c,\left(B,B\right)}=2.0\), and (c) \({r}_{c,\left(B,B\right)}\) = 2.5. In all cases, the cutoff distances for A–A and A–B interactions are fixed at \({r}_{c,\left(A,A\right)}={r}_{c,\left(A,B\right)}\) = 2.5.

To quantitatively analyze compositional segregation, one-dimensional segment density profiles were calculated by averaging along the lamellar normal direction. We examined the combined influence of \({\varepsilon }_{\left(A,B\right)}\) and \({r}_{c,\left(B,B\right)}\) on equilibrium morphologies of symmetric AB diblock copolymers. Figure 3 presents normalized density profiles of A and B blocks along the direction perpendicular to the lamellar interfaces for chains with N = 80. At larger \({r}_{c,\left(B,B\right)}\) and smaller \({\varepsilon }_{\left(A,B\right)}\), the systems exhibit distinct phase separation characterized by clearly defined domains. Higher \({\varepsilon }_{(A,B)}\), which corresponds to reduced incompatibility between A and B blocks, leads to broadened interfaces and weakened spatial segregation. Reducing \({r}_{c,(B,B)}\) at fixed \({\varepsilon }_{(A,B)}\) decreases the driving force for segregation, resulting in smoother and more symmetric density profiles, particularly in weak segregation regimes. These trends reflect the competition between enthalpic and entropic contributions, consistent with previous simulation findings33. To assess the influence of finite-size effects, additional simulations were conducted in a larger box size (50 \(\times\) 50 \(\times\) 50), and the corresponding morphologies and density profiles are provided in Supporting Information Fig. S3 and Fig. S4. The results confirm the same structural trends.

Fig. 3: One-dimensional segment density profiles of equilibrated AB diblock copolymers.
Fig. 3: One-dimensional segment density profiles of equilibrated AB diblock copolymers.The alternative text for this image may have been generated using AI.
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Normalized segment density profiles of AB diblock copolymers with N = 80 at varying \({\varepsilon }_{\left(A,B\right)}\) (0.5 – 0.9). All cases use \({r}_{c,\left(A,A\right)}={r}_{c,\left(A,B\right)}=2.5\), while \({r}_{c,\left(B,B\right)}\) is varied as a 1.5, b 2.0, and c 2.5. Red and blue lines represent the density of A and B segments, respectively.

Comparisons of density profiles (Figs. S5 and S6) show that longer chain length (N = 20, 50 to N = 80) sharpens interfacial regions. Shorter chains display more diffuse profiles, indicating increased interfacial fluctuations and weaker segregation. According to mean-field theory43, enhanced segregation in longer chains arises from increased incompatibility at higher molecular weights, resulting in clearly defined interfaces. This improvement in segregation and interface sharpness is consistent with previous computational findings and experimental scattering studies44.

To characterize the long-range periodic structures, we investigated the dependence of average domain spacing (D) on interfacial interaction strength (\({\varepsilon }_{\left(A,B\right)}\)) and the cutoff radius (\({r}_{c,\left(B,B\right)}\)), clarifying the influence of microscopic interaction range and chain length on large-scale morphologies of AB diblock copolymers. As shown in Fig. 4, D is defined as the mean spacing between adjacent A-rich and B-rich domains, averaged over equilibrated configurations. For all chain lengths, D decreases with increasing \({\varepsilon }_{\left(A,B\right)}\), reflecting weakened inter-block repulsion and reduced phase separation. This decrease in D becomes pronounced at \({\varepsilon }_{\left(A,B\right)}\) > 0.7, particularly for short-range interactions, suggesting a morphological transition from microphase-separated structures toward nearly homogeneous mixtures. At fixed \({\varepsilon }_{\left(A,B\right)}\), increasing \({r}_{c,\left(B,B\right)}\) results in larger domain spacing, with this effect more pronounced at larger chain lengths (N = 50, 80). At lower \({\varepsilon }_{\left(A,B\right)}\), the greater enthalpic penalty for A–B contacts favors demixing and promotes domain formation. A larger \({r}_{c,\left(B,B\right)}\) allows segments to interact over larger distances, thereby stabilizing more extended and segregated domains. The influence of \({r}_{c,\left(B,B\right)}\) becomes more significant at larger chain lengths, attributed to increased entropic penalties associated with chain reorganization. A comparison with morphological snapshots in Fig. 2, suggests that larger \(D\) corresponds to lamellar or bicontinuous structures, while smaller \(D\) is associated with interpenetrated and less ordered morphologies.

Fig. 4: Dependence of lamellar domain spacing on interfacial interaction strength and chain length.
Fig. 4: Dependence of lamellar domain spacing on interfacial interaction strength and chain length.The alternative text for this image may have been generated using AI.
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Domain spacing D of AB diblock copolymers as a function of inter-block interaction strength \({\varepsilon }_{\left(A,B\right)}\) for three different chain lengths: a N = 20, b N = 50, and c N = 80. Each curve corresponds to a distinct B–B interaction cutoff radius: \({r}_{c,\left(B,B\right)}=1.5\), 2.0, and 2.5. Error bars represent standard deviations computed from three independent simulations.

Three-dimensional interfacial morphology characterized by IMDS

To characterize the three-dimensional interfacial morphology of AB diblock copolymers, isosurface representations of the inter-material dividing surface (IMDS), defined by the isosurface where equal densities of A and B blocks. At a fixed chain length (N = 80), this analysis demonstrates how variations in interaction ranges affect the topology and connectivity of the microphase-separated structures. As shown in Fig. 5, the IMDS visualizations reveal significant changes in morphology, ranging from fragmented and highly curved structures to smoother, more coherent lamellar-like interfaces with changing interaction parameters. At shorter interaction ranges, IMDS isosurfaces display complex morphologies characterized by disconnected domains, frequent perforations, and sponge-like bicontinuous structures with high curvature, indicating considerable topological disorder and spatial fragmentation. By comparison, longer interaction ranges (larger \({r}_{c,\left(B,B\right)}\)) promote morphological coherence, transforming fragmented domains into continuous lamellae structures featuring smoother interfaces. This transition reflects enhanced cohesive interactions that reduce interfacial curvature and suppress topological defects.

Fig. 5: IMDS reveals morphology transitions across interaction parameters.
Fig. 5: IMDS reveals morphology transitions across interaction parameters.The alternative text for this image may have been generated using AI.
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Interfacial morphology of AB diblock copolymers with \(N=80\), visualized as isosurfaces at equal A/B density. Each row corresponds to a fixed a \({r}_{c,\left(B,B\right)}\) = 1.5, b 2.0, and c 2.5, respectively, where \({r}_{c,\left(A,A\right)}={r}_{c,\left(A,B\right)}=2.5\) are fixed at 2.5. Each column corresponds to a different \({\varepsilon }_{\left(A,B\right)}\) vales, ranging from 0.5 to 0.9.

Morphological analyses (Figs. S7 and S9) indicate that longer polymer chains favor the formation of interconnected and coherent domain structures. Enhanced mesoscale connectivity at larger chain lengths can be rationalized by increased conformational entropy penalties associated with interface formation, thus favoring fewer and larger interfaces45. Consequently, longer chains exhibit broader, smoother interfaces and fewer structural defects than shorter chains. This interpretation is supported by experimental observations that report fewer grain boundaries and structural irregularities in high molecular weight BCPs46.

Three-dimensional local density fields were projected onto two-dimensional slices to visualize the difference field \({\rho }_{A}-{\rho }_{B}\). The resulting colormaps (Fig. 6) show the spatial organization of domains, interfacial sharpness, and compositional gradients. At low miscibility (i.e., low \({\varepsilon }_{\left(A,B\right)}\)), systems with longer B–B interaction ranges exhibit domains with clear boundaries and smooth interfaces. This behavior arises from the balance between enthalpic inter-block repulsions and entropic effects, which favors coherent and well-separated domains. Alternatively, higher \({\varepsilon }_{\left(A,B\right)}\) increases packing frustration and curvature stress, resulting in enhanced intermixing and more fragmented morphologies with blurred and irregular boundaries, particularly under short-range cohesion (Fig. 6a). Further analysis of density maps (Figs. S8 and S10) indicates that shorter chains tend to form more fragmented and irregular morphologies, attributable to reduced conformational constraints and increased entropic freedom. These results demonstrate the importance of chain length and its interplay with inter-block repulsion and block cohesion in determining phase-separated morphology.

Fig. 6: Two-dimensional projection maps of segmental density contrast in AB diblock copolymers.
Fig. 6: Two-dimensional projection maps of segmental density contrast in AB diblock copolymers.The alternative text for this image may have been generated using AI.
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Normalized density contrast \({{\rm{\rho }}}_{A}-{{\rm{\rho }}}_{B}\) is projected along the z-axis of the simulation box for chain length N = 80. a–c Correspond to increasing B–B cutoff radius values: \({r}_{c,\left(B,B\right)}\) = 1.5, 2.0, 2.5, respectively, and columns represent increasing interfacial interaction strength \({{\rm{\varepsilon }}}_{\left(A,B\right)}=0.5-0.9\).

Quantitative characterization of microphase ordering

We examined the dependence of the three-dimensional structure factor peak \(S\left({q}^{* }\right)\) on the interaction strength (\({\varepsilon }_{\left(A,B\right)}\)) and the interaction range (\({r}_{c,\left(B,B\right)}\)), to evaluate how these parameters influence compositional ordering in phase-separated diblock copolymers. As shown in Fig. 7(a–c),S(q*), defined as the maximum intensity of the static structure factor calculated from the three-dimensional density field at steady state, reflects the periodicity and density contrast between A-rich and B-rich domains. For all chain lengths (N = 20, 50, 80), S(q*) generally decreases with increasing \({\varepsilon }_{\left(A,B\right)}\), indicating a gradual reduction in long-range compositional order as A–B compatibility is enhanced, and thereby suppressing phase separation. At low \({\varepsilon }_{\left(A,B\right)}\), the systems show strong microphase separation with clear domain boundaries, leading to large structure factor peaks. As \({\varepsilon }_{\left(A,B\right)}\) exceeds approximately 0.7, S(q*) decreases sharply, which is consistent with a transition to a disordered or weakly segregated state. At each value of \({\varepsilon }_{\left(A,B\right)}\), the peak magnitude is increased by larger \({r}_{c,\left(B,B\right)}\), particularly for longer chains. This enhancement is attributed to the stabilizing effect of long-range homotypic interactions, which promote domain coherence and reduce interfacial roughness. These findings are consistent with previous observations of smoother and more extended domain structures at larger \({r}_{c,\left(B,B\right)}\). Previous visualizations (i.e., Figs. 2 and 3) confirm that high \(S\left({q}^{* }\right)\) corresponds to coherent lamellar or bicontinuous morphologies, while low \(S\left({q}^{* }\right)\) is associated with disordered or mixed structures. As \({\varepsilon }_{\left(A,B\right)}\) rises, the incompatibility between A and B segments is reduced, resulting in less pronounced periodic ordering due to more interfacial mixing. By comparison, expanding \({r}_{c,\left(B,B\right)}\) and employing longer chains both contribute to the development of highly ordered domains and enhanced compositional contrast. To further analyze the morphological evolution and local packing induced by \({r}_{c,\left(B,B\right)}\), we examined the corresponding morphologies, radial distribution functions (RDF), and structural descriptors in Supporting Information Fig. S11.

Fig. 7: Peak intensity and correlation length extracted from structure factor analysis.
Fig. 7: Peak intensity and correlation length extracted from structure factor analysis.The alternative text for this image may have been generated using AI.
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Structure factor peak intensity \(S({q}^{* })\) for chain lengths (a–c); and corresponding correlation length \({\xi }_{\text{fit}}\) extracted from the low-q region of the structure factor, shown in (d–f), respectively. All results are plotted as a function of the inter-block interaction strength \({\varepsilon }_{(A,B)}\), with curves representing different B–B interaction cutoff distances: \({r}_{c,(B,B)}\) = 1.5, 2.0, and 2.5. Each data point represents an average over three independent configurations, with error bars denoting the standard deviation.

To quantitatively assess long-range structural order, we determined the correlation length \({\xi }_{\text{fit}}\), by fitting the low-q region of the three-dimensional spatial autocorrelation function. Figure 7d–f shows the variation of \({\xi }_{\text{fit}}\) as a function of \({\varepsilon }_{(A,B)}\), N, and \({r}_{c,(B,B)}\). In all systems, \({\xi }_{\text{fit}}\) decreases monotonically with increasing \({\varepsilon }_{\left(A,B\right)}\). This trend indicates that enhanced A–B attraction (corresponding to weaker incompatibility) reduces segregation, which increases curvature stress and chain stretching penalties, ultimately disrupting long-range structural coherence. This decrease is most pronounced in systems with short chains and weak B–B cohesion, where reduced packing cooperativity leads to rapid domain fragmentation and increased localization. In contrast, systems with longer chains (\(N=80\)) and greater B–B interaction range (\({r}_{c,(B,B)}=2.5\)) retain higher \({\xi }_{\text{fit}}\) values. This result suggests that chain conformational flexibility and cohesive stabilization together promote the formation of extended microphase domains.

Discussion

The Flory–Huggins parameter (χ) was further examined to quantify the extent of phase separation and compositional contrast in lamellar-forming AB diblock copolymers. Fig. S12 shows the dependence of χ on the interfacial interaction strength \({\varepsilon }_{(A,B)}\) for three representative chain lengths, with each curve corresponding to a distinct B–B interaction cutoff radius. The results indicate that χ increases with both chain length and B–B interaction range, consistent with the formation of sharper and more stable lamellar domains as shown in Figs. 2–7. As \({\varepsilon }_{(A,B)}\) increases, χ systematically decreases, reflecting enhanced A–B mixing and reduced compositional segregation. This trend is consistent with the observed decreases in the structure factor peak intensity \(S({q}^{* })\) and domain spacing D (see Figs. 4 and 7). Systems with larger \({r}_{c,(B,B)}\) and longer chains exhibit higher χ values, underscoring the combined effects of molecular architecture and interaction range in stabilizing ordered microphase morphologies. These results confirm that χ, as calculated from microscopic pairwise interaction potentials using the mean-field approximation (Eq. 8), provides a consistent scalar metric for quantifying compositional order in lamellar phases. When considered together with other quantitative descriptors such as \(S({q}^{* })\), D, and \({\xi }_{\text{fit}}\), χ offers additional insight into phase stability mechanisms and the tunability of block copolymer self-assembly.

The relationships among key thermodynamic and morphological parameters were evaluated using the Pearson correlation matrix (Fig. 8a) and a feature importance analysis with respect to domain spacing D (Fig. 8b). The correlation matrix shows strong positive correlations among three main structural descriptors: correlation length \(\left({\xi }_{\text{fit}}\right)\), peak of structure factor \(S({q}^{* })\), and domain spacing D represent the degree of long-range spatial order in the system. As expected, \({\varepsilon }_{(A,B)}\) exhibits strong negative correlations with these ordering parameters. This result confirms that \({\varepsilon }_{(A,B)}\) is a principal factor governing microphase behavior, with higher values tending to disrupt structural order unless compensated by increased chain length or enhanced cohesive interaction range. Most other parameter pairs exhibit weak to moderate correlations, suggesting that each structural descriptor captures a different aspect of microphase behavior, ranging from global periodicity to interfacial sharpness and domain connectivity. The feature importance analysis indicates that morphological stability is primarily influenced by chain length, interaction range, and segmental compatibility. These results indicate that stable morphologies are determined by the interplay among multiple parameters, highlighting the multifactorial nature of block copolymer self-assembly47.

Fig. 8: Correlation analysis and feature importance for structural descriptors contributing to lamellar domain spacing D.
Fig. 8: Correlation analysis and feature importance for structural descriptors contributing to lamellar domain spacing D.The alternative text for this image may have been generated using AI.
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a Pearson correlation coefficient between key structural metrics and microscopic parameters, including chain length N, interaction strength \({\varepsilon }_{\left(A,B\right)}\), cutoff radius \({r}_{c,\left(B,B\right)}\), structure factor features \(S({q}^{* })\), segmental order parameter (O–P), fitted correlation length \({\xi }_{\text{fit}}\), center-of-mass offset (COM), and lamellar layer count (layers). The histogram inset shows the distribution of correlation values. b Feature importance scores based on a random forest regression model, highlighting the relative contributions of different features to morphology ordering.

In addition to interaction parameters, polymer architecture is an important factor influencing morphological order. For example, ring copolymers show about 20% smaller domain spacing and more diffuse interfaces at comparable \(\chi\) attributable to reduced \(\chi\) sensitivity and increased morphological stability35. ABA triblock copolymers frequently display improved vertical alignment and sharper interfaces as a result of entropic elasticity. Although these architectures are not directly simulated in the present study, the observed trends regarding cohesive interaction range and chain extensibility may provide guidance for the design of advanced copolymer systems. The correlation and feature analyses support the use of \(S({q}^{* })\), \({\xi }_{\text{fit}}\), and D as integrated indicators of ordering stability.

In conclusion, this study systematically investigated the morphological evolution and microphase separation of symmetric AB diblock copolymers using CG-MD simulations. By independently modulating interfacial interaction strength (\({\varepsilon }_{(A,B)}\)) and cohesive interaction range (\({r}_{c,(B,B)}\)), we quantitatively analyzed the domain spacing (D), structure factor peak intensity (\(S({q}^{* })\)), and correlation length (\({\xi }_{{fit}}\)) to characterize the morphological stability and order. The results demonstrate that enhanced cohesive interactions by increasing cutoff distance (\({r}_{c,(B,B)}\) ≥ 2.0) effectively stabilize lamellar morphologies and mitigate curvature-driven interfacial instabilities, sustaining the structural regularity even at elevated \({\varepsilon }_{(A,B)}\) values. Correlation analysis further confirms the critical roles of increased chain length and extended B-block interaction range in promoting long-range structural coherence and domain clarity. These findings offer a reliable quantitative link between microscopic interaction parameters and established segregation theories, providing valuable design principles for optimizing self-assembled structures in advanced thermoplastic elastomers and functional block copolymer systems. Furthermore, although the present work focuses on symmetric diblock copolymers, the mechanistic trends and parameter dependencies revealed here are expected to offer predictive insights for modulating microphase-separated morphologies in more structurally or energetically asymmetric systems.

Methods

Overview of the CG modeling

CG-MD simulations are widely used for investigating polymer systems across extended spatial and temporal scales48. In this study, each polymer chain is modeled as a symmetric AB diblock architecture, consisting of equal-length A and B segments, with each block containing N/2 monomers (Fig. 1). Phase separation between the A and B blocks is driven by their mutual incompatibility, which is controlled by the LJ interaction parameters: the interaction strength parameter \({\varepsilon }_{(i,j)}\) and cutoff radius \({r}_{c,(i,j)}\) defined as:

$${U}_{\text{LJ}}\left(r\right)=4{\varepsilon }_{\left(i,j\right)}\left[{\left(\frac{{{\rm{\sigma }}}_{\left(i,j\right)}}{r}\right)}^{12}-{\left(\frac{{{\rm{\sigma }}}_{\left(i,j\right)}}{r}\right)}^{6}-{\left(\frac{{{\rm{\sigma }}}_{\left(i,j\right)}}{{r}_{c,\left(i,j\right)}}\right)}^{12}+{\left(\frac{{{\rm{\sigma }}}_{\left(i,j\right)}}{{r}_{c,\left(i,j\right)}}\right)}^{6}\right]$$
(1)

for \(r < {r}_{c,(i,j)}\), and \({U}_{LJ}\left(r\right)\) = 0 for \(r > {r}_{c,(i,j)}\), where r is the interparticle distance, with cutoff radius \({r}_{c}=2.5\) and σ is the length scale parameter of the interaction. Parameters \(\sigma\) and \(\varepsilon\) represent the effective van der Waals radius and potential well depth, respectively.

Simulations of symmetric AB diblock copolymers, consisting of 500 chains with a total of 20,000 beads, were conducted using the classical bead–spring model. Bonded interactions between adjacent beads were described by a harmonic potential:

$${U}_{{bond}}=\frac{1}{2}{K}_{b}{\left(r-{r}_{0}\right)}^{2}$$
(2)

Standard Lennard–Jones reduced units were used \((\sigma =1.0\), \(\varepsilon =1.0\), \(m=1.0\)), establishing natural time units \(\tau =\sqrt{\sigma m/\varepsilon }\), the pressure was set to 0 (in reduced units). The equations of motion were integrated via the velocity-Verlet algorithm with a timestep \(\,\Delta t=0.005\tau\). Temperature was maintained at \(T=0.8\) using a Langevin thermostat (damping coefficient \(\gamma =100\,{\tau }^{-1}\)), and periodic boundary conditions were applied in all directions.

All simulations were carried out using the bead–spring model as implemented in the LAMMPS molecular dynamics package49. Initial configurations were generated by uniformly distributing polymer chains in a cubic box (\(L=25,\) polymer volume fraction \(\phi \approx 1\)). Initial equilibration was performed using purely repulsive Weeks–Chandler–Andersen (WCA) interactions (LJ truncated at \({r}_{c}={2}^{1/6}\)) lasted \(5\times {10}^{6}\) steps. Microphase separation was initiated by activating attractive interactions among B-type beads, with the cutoff radius \({r}_{c,(B,B)}\) was varied from \(1.5\) to \(2.5\). The interaction strengths were set to \({\varepsilon }_{(A,A)}={\varepsilon }_{(B,B)}=1.0\), while A–B interfacial interaction strength \({\varepsilon }_{(A,B)}\) varied from 0.5 to 0.9. Cutoff radii for A–A and A–B interactions were kept constant at \({r}_{c,(A,A)}={r}_{c,(A,B)}=2.5\). Production simulations were performed in an anisotropic NPT ensemble using Nosé–Hoover thermostat and barostat, allowing box fluctuations along the x-axis to facilitate lamellar formation. Each production run consisted of \(5\times {10}^{7}\) steps, including \(1\times {10}^{7}\) steps for re-equilibration and \(4\times {10}^{7}\) steps for structural characterization. Snapshots were collected every \(1\times {10}^{5}\) steps.

To relate reduced units to real units, for example, a standard Kremer–Grest mapping with \(\varepsilon \sim 1.6\) kcal mol−1 and \(\sigma \sim 0.5\) nm yields a conversion factor of order \(\varepsilon /{\sigma }^{3}\sim\) \({10}^{2}\) J cm−3, which can be used to estimate trends rather than absolute values: for higher \({r}_{c}=2.5\), the cohesive energy density (CED) is \(\sim\) 80–100 J cm−3 with \({T}_{g}\sim 370\) K (\({T}_{g}^{* }\) ~ 0.45), resembling a polystyrene (PS)-like (high-\({T}_{g}\)) polymer; for lower \({r}_{c}=1.5\), the CED is \(\sim 60-80\) J cm−3 and \({T}_{g}\sim 300\) K (i.e., \({T}_{g}^{* } \sim 0.37\) in reduced units), corresponding to a polybutadiene (PB)-like (low-\({T}_{g}\)) polymer. It should be noted that the experimental CEDs of polymers (e.g., PS \(\sim 340-350\) J cm−3) are usually higher than these CG estimates due to the reduced degree of freedom and a lack of atomic-level interactions in CG modeling. Thus, such mapping should be interpreted as qualitative guidance.

Morphology characterization

One-dimensional density profile

The segment density profile of the A-block, \({\rho }_{A}\left(z\right)\), was calculated along the lamellar normal (z-direction) by dividing the simulation box into \({N}_{z},\) each with thickness \(\Delta z={L}_{z}/{N}_{z}\). The local density in each slab was determined as:

$${\rho }_{A}\left(z\right)=\frac{1}{A\Delta z}\sum _{i\in A}\Theta \left({z}_{i}-z+\frac{\Delta z}{2}\right)\Theta \left(z+\frac{\Delta z}{2}-{z}_{i}\right)$$
(3)

where\(A={L}_{x}{L}_{y}\) is the cross-sectional area, \({z}_{i}\) is the z-coordinate of the ith A-type monomer, and \(\Theta\) denotes the Heaviside step function. The density profile was averaged over the production trajectory.

Domain spacing calculation

The lamellar domain spacing, D, was determined from the periodicity of the one-dimensional density profile50. Specifically, the positions of adjacent maxima in \({\rho }_{A}\left(z\right)\) were identified, and the mean distance between neighboring maxima was calculated as \(D={\rm{\langle }}{z}_{i+1}-{z}_{i}{\rm{\rangle }}\), where \({z}_{i}\) and \({z}_{i+1}\) denote the positions of consecutive peaks in the density profile. The value of D was averaged over all identified peak pairs in the simulation cell and over all sampled configurations.

Three-dimensional structure factor

The three-dimensional structure factor51,52, \(S({q}^{* })\), was calculated as the Fourier transform of the local composition field, \(\psi \left(r\right)={\rho }_{A}\left(r\right)-{\rho }_{B}\left(r\right)\):

$$S\left(q\right)=\frac{1}{V}{\left|{\int }_{V}{\rm{\psi }}\left(r\right){e}^{-{iq}\cdot r}{dr}\right|}^{2}$$
(4)

where \({\rho }_{A}\left(r\right)\) and \({\rho }_{B}\left(r\right)\) are the local densities of A and B segments, respectively. Where V is the system volume. In practice, \(\psi \left(r\right)\) was discretized onto a cubic grid and transformed using the Fast Fourier Transform (FFT). The spherically averaged structure factor, \(S({q}^{* })\), was obtained by averaging \(S\left(q\right)\) over all wavevectors of magnitude \(q=\left|q\right|\).

Correlation length determination

The spatial correlation length53,54, \(\xi\), was extracted from the real-space autocorrelation function of the composition field, defined as

$$C\left(r\right)=\frac{\left\langle {\rm{\psi }}\left({r}_{0}\right){\rm{\psi }}\left({r}_{0}+r\right)\right\rangle }{\left\langle {{\rm{\psi }}}^{2}\right\rangle }$$
(5)

where the brackets denote spatial averaging. The autocorrelation was computed by inverse FFT of \({\left|\hat{{\rm{\psi }}}\left(q\right)\right|}^{2}\) and radially averaged. The short-range decay of C(r) was fitted to an exponential function, \(C\left(r\right)\sim \exp \left(-r/\xi \right)\).

Flory–Huggins parameter estimation

The Flory–Huggins interaction parameter (\(\chi\)) was estimated from the microscopic interaction parameters (\({\varepsilon }_{(A,B)}\), \({r}_{c,(B,B)}\)) by applying a mean-field approximation for binary mixtures55. The relationship between \(\chi\) and the microscopic interaction potentials is given by

$$\chi =\frac{{\rm{\rho }}}{{k}_{B}T}{\int }_{0}^{\infty }\left[2{U}_{{AB}}\left(r\right)-{U}_{{AA}}\left(r\right)-{U}_{{BB}}\left(r\right)\right]{g}_{{AB}}\left(r\right)2{\rm{\pi }}{r}^{2}{dr}$$
(6)

where \({U}_{i,j}\left(r\right)\) is the pairwise interaction potential between i and j, \({g}_{{AB}}\left(r\right)\) is the RDF, \(\rho\) is the number density, and \({k}_{B}T\) is the thermal energy. If the cutoff lengths (\({r}_{c}\)) and particle sizes are the same for all pairs, the Eq. 6 can be rewritten as

$$\chi =\frac{{\rm{\rho }}}{{k}_{B}T}{\int }_{0}^{{r}_{c}}\left[2{{\rm{\varepsilon }}}_{{AB}}-{{\rm{\varepsilon }}}_{{AA}}-{{\rm{\varepsilon }}}_{{BB}}\right]U\left(r\right){g}_{{AB}}\left(r\right)2{\rm{\pi }}{r}^{2}{dr}$$
(7)

The above equation can be even simplified with functional form of intermolecular interaction is given,

$$\chi =\frac{{\rm{\rho }}}{{k}_{B}T}{\int }_{0}^{{r}_{c}}\left[2{{\rm{\varepsilon }}}_{{AB}}{U}_{{LJ},{AB}}\left(r\right)-{{\rm{\varepsilon }}}_{{AA}}{U}_{{LJ},{AA}}\left(r\right)-{{\rm{\varepsilon }}}_{{BB}}{U}_{{LJ},{BB}}\left(r\right)\right]\exp \left[-{U}_{{LJ},{AB}}\left(r\right)\right]2{\rm{\pi }}{r}^{2}{dr}$$
(8)

where all terms are evaluated using the appropriate LJ parameters for each pair50. For example, \({U}_{{LJ},{AB}}\left(r\right)\) refers to the pair-wised interaction energy between bead A and B based on Eq. 1.