United-Atom Molecular Dynamics Simulations of Strain-Induced Crystallization in Polyisoprene Melts

Abstract

Stretching polyisoprene (PI) beyond a certain strain level causes strain-induced crystallization, resulting in nanocrystallite formation and toughening of the material. Although scattering experiments have confirmed the existence of nanocrystallites, their distribution and spontaneous formation remain unclear. Here, we present united-atom molecular dynamics (UA-MD) simulations that capture the spontaneous formation of nanocrystallites in stretched PI melts from an oriented state. Using molecular-level snapshots based on our PI-crystal local order parameter and two-dimensional scattering patterns, we detect the grown nanocrystallites. We further show that tiny nanocrystallites formed during rapid stretching can either grow or disappear during the accessible time scale of the subsequent isothermal crystallization stage. Importantly, our simulations demonstrate that the threshold for nanocrystallite growth correlates better with the initial segmental orientation than with the strain level alone. Our UA-MD approach provides molecular-level insights relevant to nanocrystallite formation and self-reinforcement in cross-linked PI networks, offering guidance for the design of tough soft materials.

Strain-induced crystallization (SIC) has been widely explored as a strategy for toughening various soft materials. − For instance, natural rubber, which is primarily composed of cis-1,4-polyisoprene (PI), exhibits SIC when stretched beyond a certain strain; this phenomenon induces self-reinforcement via the formation of nanocrystallites, which act as effective cross-link-like reinforcing points. Recently, Mayumi et al. − developed tough soft materials by implementing SIC. Over the years, scattering experiments (including those performed in the aforementioned studies) have confirmed the existence of such strain-induced nanocrystallites, which were first reported by Katz a century ago. Even in the absence of cross-links, such as in PI melts, SIC was observed when strain was applied via rapid deformation. Flory highlighted the importance of a temporary segment arrangement as a precursor to crystal growth; crucially, he pointed out that practically observed nanocrystallites in the SIC of PI represent kinetic phenomena, unlike ideal thermodynamically stable crystallization at equilibrium. Notably, these kinetically formed nanocrystallites persist even above the nominal melting temperature T _m, playing a crucial role in the reinforcement effect. To date, however, the molecular-level behaviors of these kinetically formed nanocrystallites remain unclear.

Molecular dynamics (MD) simulations of a united-atom (UA) model are expected to be a powerful tool for elucidating the spontaneous formation behavior of nanocrystallites. The UA model was developed based on the fundamental idea of uniting hydrogen and carbon atoms to maintain the carbon backbone structure, which made the simulations more efficient in terms of computation costs compared to all-atom MD simulations. In the case of polyethylene (PE), simulations have been conducted to investigate the SIC of stretched PE − and flow-induced crystallization (FIC) − of PE under steady flows. To investigate thermodynamically stable crystallization, the FIC protocol was designed to achieve a steady-state extensional flow in the melt, keeping the Deborah number constant even upon quenching to a supercooled state. For PI, however, performing such a protocol is highly challenging because the intrinsic dynamics of PI are significantly slower than those of PE. Prior to investigating thermodynamic stability or nucleation kinetics, demonstrating the UA-MD simulations of the spontaneous packing behavior of stretched PI melts remains a crucial milestone. Therefore, the present simulations focus on the formation and early stage evolution of nanocrystallites within the accessible simulation time, rather than on establishing their thermodynamic stability.

In this study, the SIC behavior of stretched melts with long PI chains was successfully observed by conducting large-scale UA-MD simulations. The spontaneous growth or disappearance of tiny nanocrystallites formed during the initial stretch was investigated during isothermal crystallization (IC) simulations. The sophisticated UA potential provided reasonable reproducibility of physical properties, and high computing efficiency was achieved owing to the omission of long-range electrostatic interactions. By applying uniaxial deformation via extensional flow in the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) MD program, , we efficiently generated a systematic set of PI melt configurations at various strain levels (λ), each of which originated from the same equilibrated state for each independent initial configuration. To clarify the dependence of the crystallization behavior on the chain length, we also investigated PI melts with different numbers of isoprene units per chain (n _PI = 48, 96, 192, 384, and 768). All simulations were performed at T = 300 K, which is close to the nominal T _m of PI. In experimental studies − investigating kinetically driven SIC behaviors, performing crystallization near the T _m is crucial to suppress the interference of thermodynamically stable crystallization. Furthermore, nanocrystallites formed via SIC exhibit remarkable thermal stability, resisting melting even well above the T _m. Therefore, a temperature of 300 K was considered suitable to observe the strain dependence of the nanocrystallite formation process based on the phase diagram for λ and T, which was obtained via nuclear magnetic resonance and wide-angle X-ray diffraction (WAXD) experiments.

We prepared initial configurations using the multiscale approach (), without quenching. The obtained initial configurations were reasonable because the kink-based entanglement length was approximately 28 isoprene units. While the accessible simulation time is much shorter than the experimental entanglement time (on the order of 10 μs), , it provides access to the early stage formation and evolution of nanocrystallites under rapid stretching, as discussed later. For the multiple independent configurations, rapid stretching with a strain rate of 2.5 × 10^–8 fs^–1 was applied. The deformation was stopped immediately after each target λ was achieved. Subsequently, for each λ, multiple independent IC simulations were performed under constant NVT conditions, without further deformation, as described in the .

Figure shows a snapshot of a PI melt (n _PI = 192) stretched at λ = 5, which exhibits a partially crystallized structure formed during SIC. This was obtained by IC simulations for an IC time t _IC = 500 ns after uniaxial stretching. In the snapshot, the color of the UA atom is based on the local order parameter (P _P) of the PI crystals. Here, the P _P,i of the ith isoprene unit is defined as

$$P_{\mathrm{P},i}=\frac{1}{n_{\mathrm{Nei},i}}\left(\sum _{j=1}^{n_{\mathrm{Nei},i}}\frac{3{({\mathit{n}}_i^{\mathrm{P}}·{\mathit{n}}_j^{\mathrm{P}})}^2-1}{2}\right)$$ 1

where n _i ^P is the unit vector perpendicular to the plane of the ith isoprene unit, and n _Nei,i denotes the number of neighboring C= atoms within the cutoff length, r _c‑P of 0.57 nm. The hyperparameters, such as r _c‑P and P _P,c, are discussed in the Supporting Information. P _P,c = 0.85 is the threshold to distinguish between crystalline and amorphous domains for each isoprene unit. Remarkably, the expected sizes of the single nanocrystallites aligned with the experimental observations presented in Figure B. As shown in Figure A and , the PI melts formed nanocrystallites along the stretching (c) direction (Figure C–E). Further, Figures S8–S22 show the snapshots of the PI melts that underwent SIC under all examined cases of n _PI and λ.

1.

Snapshot of the strain-induced crystallization (SIC) of a polyisoprene (PI) melt. (A) Snapshot of a PI melt with n _PI = 192 stretched at λ = 5. The presented square box is sized 24 nm × 24 nm, and the slice thickness of the crystallized melt is 1 nm. OVITO was used for visualizing the data. Three-dimensional (3D) views are provided in the . (B) Schematic of the dimensions of the nanocrystallite relative to the same axes as in (A). The dimension in the stretching direction is almost independent of the strain and temperature. (C–E) Crystal structures along the a-, b-, and c-axes, respectively. The unit cell parameters were finely determined by Rajkumar et al.

Figure A shows a 3D visualization of a stretched PI melt before it underwent SIC (t _IC = 0). Figure B–D show the progression of the SIC of the PI melt. By comparing Figure A (t _IC = 0) with Figure B (t _IC = 500 ns), we observed the growth of nanocrystallites with atom packing; this result provides direct molecular evidence that complements previous speculations based on experimental studies. A comparison of the results at λ = 5 (Figure B) and λ = 11 (Figure C) revealed that the number of nanocrystallites increased with λ. The snapshot of the PI melt with n _PI = 768 (Figure D) stretched at λ = 5 indicates that the number of nanocrystallites increased with n _PI. This packing phenomenon was further evidenced by the changes in the scattering pattern. A single chain observation (Figures E and S23–S25) indicates the coexistence of bundle-like and folded local crystallizations, which has been discussed for many years since the study by Flory. Locally bundled assemblies of highly oriented partial chains did not necessarily crystallize immediately; instead, crystallization tended to proceed via local rearrangement. The possibility of observing fully folded lamellar crystallites through long-term simulations with skillfully controlled conditions, such as the temperature and chain length, was not ruled out. In general, locally formed nanocrystallites are expected to act as effective cross-linking points and serve as the origin of the toughness of the material.

2.

3D visualization of the SIC of PI melts. The n _PI, λ, and t _IC were (A) 192, 5, and 0 ns; (B) 192, 5, and 500 ns; (C) 192, 11, and 500 ns; and (D) 768, 5, and 1500 ns, respectively. (E) A snapshot of a single chain with n _PI = 768 under λ = 5 at t _IC = 0, 500, and 1500 ns. The colors are based on the local order parameters (P _P) shown in Figure .

Figure A–C shows a series of two-dimensional scattering patterns (2DSPs) from strain-induced orientations to the SIC of a PI melt with n _PI = 192 at t _IC = 0, 200, and 500 ns. These 2DSPs were calculated using a circular average of 3D Fourier transformations of the atomic density fields. ,,, Statistical averages in the present work were obtained from five independent runs. The sharpness of the scattering peak positions increased with the t _IC during IC simulations under stretching, which is similar to the behavior observed in WAXD experiments. − Notably, the weak spots at (201) and (002) in Figure A indicate the presence of tiny nanocrystallites or ordered precursors that emerged during rapid stretching. In the subsequent IC simulations, we examined whether these initially formed nanocrystallites grow or disappear under the influence of local thermal segmental relaxation.

3.

Two-dimensional scattering patterns and the results of peak spot analysis. (A–C) 2DSPs for PI melts with n _PI = 192 stretched at λ = 5 after t _IC = 0, 200, and 500 ns, respectively. (D, E) Intensity of the (201) spot for the melts with n _PI = 192 and 768, respectively, stretched at λ = 5. (F) Phase diagram of the SIC behavior in terms of λ and n _PI based on the 2DSPs obtained at t _IC = 100 ns for n _PI = 48, 500 ns for n _PI = 96, 192, and 384, and 1500 ns for n _PI = 768, respectively.

Figure D,E illustrate the intensity distributions of the (201) spots of the PI melts with n _PI = 192 and 768, respectively, stretched at λ = 5. We selected the (201) spot from the four major WAXD spots ((200), (120), (201), and (002)) of the crystallized PI because the bright (120) spot was too close to the spot of chain orientation caused by the rapid stretching, and the relative weakness of the (200) and (002) spots indicated that the growth in the (200) and (002) directions could evolve at a later stage of crystallization, e.g., at t _IC on the order of milliseconds, as observed in a recent WAXD study. The 2DSPs for other cases are provided in Figures S26–S32. In contrast to the results for the PI melts with n _PI = 192 and 768 stretched at λ = 5 (Figures F, S29, and S32), no growth of the (201) spot was observed for the cases with n _PI = 96 and 192 stretched at λ = 4 (Figures F, S27, and S29). Instead, their 2DSPs became more diffuse, approaching a ring pattern with increasing t _IC (Figure F). This result is consistent with the disappearance of the initially ordered precursors. Furthermore, the 2DSPs in Figure F clearly show that the degree of crystallization increases with λ. The threshold at λ = 5 is consistent with previously reported experimental results. ,,

Figure A–C shows the snapshots of the crystallized PI melts perpendicular to the stretching direction for (n _PI, λ) = (192, 5), (192, 11), and (768, 5), respectively. The cross-sectional areas of single crystallites increased with λ and n _PI, reflecting the crystallinity, χ, defined as the fraction of isoprene units classified as crystalline by the local order parameter P _P. The snapshots reveal an increase in both the number and the apparent size of the nanocrystallites at λ = 11 compared with those at λ = 5. Furthermore, topological data analysis (Figures S33–S36) indicates the presence of nanocrystallite aggregates at λ = 11. Figure S37 reveals that the average cluster size increased with the t _IC for the cases presented in Figure A–C. Figure S38 clarifies that the nanocrystallites of the PI melt with n _PI = 768 disappeared at λ = 3 but increased in size at λ = 4. Importantly, these results demonstrate that tiny nanocrystallites formed during the initial rapid stretching are not guaranteed to grow under the present rapid stretching protocol; instead, they can disappear during IC when the imposed stretch ratio is below the growth threshold. This behavior suggests the existence of a critical size for sustained nanocrystallite growth at a given temperature. Regarding PE crystal growth, Rutledge and co-workers − have systematically evaluated the critical nucleus size. Building on this approach, evaluating such a critical size in stretched PI melts is desirable, though it remains a future challenge owing to the slower crystallization process of PI and the high computational cost.

4.

Snapshots of the SIC of PI melts perpendicular to the stretching direction and their crystallinities, χ. In (A), (B), and (C), the stretching direction is normal to the viewing plane. Snapshots corresponding to n _PI, λ, and t _IC of (A) 192, 5, and 500 ns; (B) 192, 11, and 500 ns; (C) 768, 5, and 1500 ns, respectively. (D) Dependence of χ on t _IC at n _PI = 192. (E) Dependence of χ on λ and n _PI.. (F) Plot of χ vs the Hermans orientation factor, f _H, at t _IC = 0. Here, the values of χ were obtained at t _IC = 100 ns for n _PI = 48, 500 ns for n _PI = 96, 192, and 384, and 1500 ns for n _PI = 768, respectively.

Figure D shows the t _IC dependence of χ for melts with n _PI = 192. At λ < 5, χ decreases with increasing t _IC, indicating that the tiny nanocrystallites formed during the initial rapid stretching disappear during IC. In contrast, at λ ≥ 5, χ increases with increasing t _IC, and we observe that the nanocrystallites grow spontaneously. At larger t _IC values in the present simulations, we can distinguish between the spontaneous growth and disappearance of the initially formed nanocrystallites. Notably, we also confirmed that the crystallinity approximately correlates with the fraction of trans conformations for single-bonded carbons (Figure S39), as indicated by Flory. Figure S40 reveals that the conformation within the crystal regions was almost entirely trans.

Figure E shows χ as a function of λ for each n _PI. The thresholds of λ and n _PI are consistent with the phase diagram shown in Figure F. We found that the n _PI dependence of χ was small for large n _PI values (n _PI = 192, 384, and 768) but more pronounced for small n _PI values (n _PI = 48 and 96). Therefore, a more relevant descriptor independent of n _PI is highly desirable. As predicted by Flory, a precursor to SIC behavior caused by the strong initial stretching is reflected in the high orientational order, which is captured by the Hermans orientation factor, f _H in .

Figure F shows χ against the Hermans orientation factor immediately after rapid stretching, f _H(t _IC = 0). The relationship between f _H(t _IC = 0) and χ showed little dependence on n _PI, suggesting that f _H(t _IC = 0) can be regarded as a better descriptor similar to the flow-induced nematic precursors observed in PE crystal nucleation. − Furthermore, we emphasize the importance of segmental orientation as a key determinant of crystallization kinetics, as discussed for PI (natural rubber). ,

For the shortest chain (n _PI = 48), Figure F indicated that the lower χ can be largely explained by its inability to achieve a high f _H(t _IC = 0) even at a high λ. Although shorter chains suffer from an effective T _m depression via the Flory–Fox relation, the effect of this T _m depression played a minor role in the subsequent IC stage, which was instead predominantly governed by the established f _H(t _IC = 0). This T _m depression effect is considered to be inherently factored into the lower initial f _H(t _IC = 0) itself.

Based on this understanding of orientational importance, in the stretched PI melts without cross-linking under the present rapid stretching condition, we infer that entanglements played a major role in the SIC within the time scale accessible to the present UA-MD simulations. Because the Deborah number was sufficiently large on the time scales studied here, entanglements likely acted as quasi-cross-links. At longer, experimentally relevant time scales, where entanglement constraints may relax, a more direct correlation with experimental results will require both substantially longer simulations and explicit studies of cross-linked PI.

In this study, we successfully used UA-MD simulations to reproduce the fundamental structural features of the SIC in PI melts at a level comparable to that observed in scattering experiments. The results were validated at the macroscopic level, and the time evolution of the simulated 2DSPs qualitatively reproduced key features observed in previous WAXD experiments. − Additionally, at the microscopic level, we defined a local order parameter P _P to distinguish PI crystalline domains, which enabled us to obtain snapshots that clearly revealed the spatial distribution of the nanocrystallites formed during SIC. Using these analyses, we further examined the structural evolution of tiny nanocrystallites formed during rapid stretching, specifically focusing on whether the nanocrystallites grow or disappear during IC. Consequently, we identified a threshold-like condition for the growth of nanocrystallites within the present simulation time, which correlated better with the initial orientational order f _H(t _IC = 0) than with the macroscopic strain λ alone. These findings demonstrate that UA-MD is an effective approach for studying crystallization in PI melts. The study findings provide a basis for future molecular simulation studies of PI and natural rubber.

Furthermore, the success of the UA-MD simulations of PI suggests that the carbon backbones of polymers play a crucial role in localized crystallization. The extension of the present framework to other semicrystalline polymers and cross-linked PI networks will be an important next step. The integration of experiments and simulations for analyzing SIC behaviors in various polymers is a promising direction for the molecular-level development of tough soft materials.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsmacrolett.6c00036.

• Movie S1 (MP4)

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• Movie S3 (MP4)

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• Movie S5 (MP4)

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• Movie S8 (MP4)

• Movie S9 (MP4)

• Methods; Development of crystal local order parameters; Snapshots; Single chain observations; Two-dimensional scattering patterns; Topological data analysis; trans-Conformation analysis; Hermans orientation factor; Detailed supporting movie descriptions (PDF)

The manuscript was written with contributions from all authors. The conceptualization, computations, analyses, and visualizations were mainly performed by K.H. All authors approved the final version of the manuscript.

The authors declare no competing financial interest.