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. 2021 Dec 20;25(1):103674. doi: 10.1016/j.isci.2021.103674

Quantum tunneling of hydrogen atom transfer affects mandrel degradation in inertial confinement fusion target fabrication

Yu Zhu 1, Xinrui Yang 1, Famin Yu 1, Rui Wang 1, Qiang Chen 2, Zhanwen Zhang 2,∗, Zhigang Wang 1,3,∗∗
PMCID: PMC8733146  PMID: 35024593

Summary

Poly-α-methylstyrene (PAMS) is considered as the preferred mandrel material, whose degradation is crucial for the fabrication of high-quality inertial confinement fusion (ICF) targets. Herein, we reveal that hydrogen atom transfer (HAT) during PAMS degradation, which is usually attributed to the thermal effect, unexpectedly exhibits a strong high-temperature tunneling effect. Specifically, although the energy barrier of the HAT reaction is only 10−2 magnitude different from depolymerization, the tunneling probability of the former can be 14–32 orders of magnitude greater than that of the latter. Furthermore, chain scission following HAT will lead to a variety of products other than monomers. Our work highlights that quantum tunneling may be an important source of uncertainty in PAMS degradation, which will provide a direction for the further development of key technology of target fabricating in ICF research and even the solution of plastic pollution.

Subject areas: Theoretical physics, Quantum physics, Nanotechnology fabrication

Graphical abstract

graphic file with name fx1.jpg

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Highlights

  • •

    Tunneling of hydrogen atom transfer (HAT) brings uncertainty to mandrel degradation

  • •

    Lower energy barrier and stronger tunneling make active-end HAT occur more easily

  • •

    Chain scission following HAT leads to a variety of products other than monomers

Theoretical physics; Quantum physics; Nanotechnology fabrication

Introduction

Inertial confinement fusion (ICF) (Betti and Hurricane, 2016; Atzeni and Meyer-ter-Vehn, 2004; Lindl, 1998), one of the most promising ways to control thermonuclear fusion reaction, has shown huge potential for the ultimate realization of fusion ignition (Jeff, 2021; Zhang et al., 2020; Edwards et al., 2013). Among the multiple parts of ICF experiments, target is essential, whose quality has an extremely important influence on the reliability and degree of precision for subsequent experimental results (Tucker-Schwartz et al., 2010; Antipa et al., 2013). In recent years, the degradable mandrel technique with poly-α-methylstyrene (PAMS) degradation as the core has become one of the main technologies for fabricating high-quality ICF targets (Letts et al., 1995; Fearon et al., 1997). For this technology, one key problem is how to achieve effective degradation of PAMS. Despite the continuous improvement of the synthesis methods and experimental conditions, the degradation of PAMS, just like the degradation of synthetic plastics, cannot completely occur in accordance with the ideal depolymerization reaction to produce monomer, which brings about the uncertainty of degradation (Brown and Wall, 1958; Poutsma, 2007). Obviously, this issue is not only significant for ICF research but also fundamental to the understanding of mechanisms of environmental pollution caused by common plastics that are also composed of hydrogen and carbon, etc.

Notably, researchers have realized that hydrogen atom transfer (HAT) is a universal reaction during degradation, which will lead to changes in the polymer structure. It has been well accepted that H atom on the main chain of polymers composed of light elements can be seized by free radical to form a new intrachain radical which will cause random chain scission (Poutsma, 2007; Guyot, 1986). In this view, the occurrence of HAT reaction in PAMS may be an important reason that leads to the degradation of PAMS deviating from the ideal depolymerization process. However, although experimental technology (e.g., in situ measurements of spectra) has been greatly improved to study the reaction mechanism (Kalampounias et al., 2003), direct experimental descriptions are still lacking. Moreover, for such a long-distance, HAT process that involves breaking the chemical bond or even overcoming the steric hindrance effect, it is not relatively easy to achieve from the perspective of classical thermal disturbance. Therefore, it is necessary to conduct research on HAT reactions in PAMS at the atomic level based on a new perspective.

Actually, owing to the light mass of the H atom, many processes involving it are susceptible to quantum-mechanical effects, such as tunneling (Bell, 1980; McMahon, 2003; Schreiner, 2017; Mayer, 2011; Meisner and Kästner, 2016). Tunneling by H species is of critical importance for many reactions, such as proton-coupled electron transfer reactions (Hammes-Schiffer, 2001; Glover et al., 2017), redox enzyme reactions (Knapp and Klinman, 2002; Gao and Truhlar, 2002), conformational inversion of molecules (Brunton et al., 1976; Schreiner et al., 2011; Mardyukov et al., 2017; Koch et al., 2017), and rotation of hydrogen bonds (Ranea et al., 2004; Richardson et al., 2016). Generally, tunneling is significant at low temperatures, although H tunneling is not restricted to such temperatures (McMahon, 2003; Schreiner et al., 2011; Mardyukov et al., 2017). These attracted our attention: for PAMS that degrades at temperatures above room temperature, can the tunneling effect in their HAT play a leading role under this temperature condition?

Based on the above, we focus on the HAT reactions in PAMS to study their classical over-barrier process and through-barrier quantum tunneling process based on the high-precision quantum mechanics calculations. Surprisingly, the results show that the strong high-temperature tunneling effect presented in HAT is significant for degradation, which may be an important source of the complexity for degradation problems. These findings contribute to achieving more efficient degradation of PAMS for the fabrication of ICF targets and can provide useful guidance for further understanding the plastic degradation.

Results and discussion

Reaction paths of hydrogen atom transfer and depolymerization

Considering the reliability and resource requirements of quantum calculations, the di-radical PAMS tetramer was used, for which the rationality of the structure selection has been confirmed in our previous studies (Zhu et al., 2021; Yu et al., 2015). First, we obtained the possible reaction paths by means of TS optimization and intrinsic reaction coordinate (IRC) analysis. Detailed methods introduction can be seen STAR Methods. The TSs structures and reaction paths can be seen in Figures S1 and S2. Their schematic diagram and corresponding potential energy surfaces (PESs) are shown in Figure 1. For the di-radical structure, there are two kinds of HAT paths at each head end (C-unsaturated end) and tail end (CH2-unsaturated end). By taking the tail end as an example, due to the steric hindrance of CH2 in the ortho position, the C atom at the tail end will seize the H atom on the methyl or methylene group of the adjacent monomer. However, because the nearby benzene ring also has steric hindrance, neither reaction is a simple one-step process. At first, this benzene ring will flip to form an intermediate (iso2). There are two cases afterward. In the first case, the C atom at the tail end will directly seize the H atom from the neighboring methyl group (Htra-ta1). In the second case, a “back-biting” process will continue to occur at the tail end, resulting in the formation of a 6-membered ring intermediate (iso3). Subsequently, the H atom from the methylene group on the second monomer (head-end monomer is the first) will be transferred to the tail end (Htra-ta2). These two kinds of HAT reactions were also observed in the dynamic simulations (Figure S3). Similarly, there are two reactions at the head end. The difference is that the H atom on the methyl group of the second monomer can be transferred directly to the head end (Htra-he1), whereas the transfer of the H atom on the methylene (Htra-he2) requires a prerequisite: the polymer must first undergo an isomerization process to form a 5-membered ring intermediate (iso1). The coordinates of the structures in the above reactions can be seen in Table S1.

Figure 1.

Figure 1

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Reaction paths of PAMS

(A) Schematic diagram of reaction paths.

Htra-he1 indicates that the H atom on the methyl group is transferred to the head end (abbreviated as “he,” C-unsaturated end). iso1-3 refers to isomerization processes. Htra-he2 represents that the H atom on the methylene is transferred to the head end after occurring iso1. Htra-ta1 indicates that the H atom on the methyl is transferred to the tail end (abbreviated as “ta,” CH2-unsaturated end) after occurring iso2. Htra-ta2 represents that the H atom on the methylene is transferred to the tail end after Int2 occurring iso3. Dhe and Dta are paths that produce the AMS monomer, which has been studied in our previous work 30.

(B) Potential energy surfaces of these nine reactions. “R,” “Int,” and “P” represent the reactant, intermediate, and product, respectively. The energy of the reactant is taken as zero. The energy values were all corrected by ZPE. The pink and light blue areas in the figure represent the reaction paths related to the head end and tail end, respectively.

Classical over-barrier and through-barrier quantum tunneling process

Next, we analyzed these reactions from the perspective of energy (Figure 1B). For three isomerization processes, including the benzene ring flip, “back-biting” of the tail end to form a 6-membered ring, and “back-biting” of the head end to form a 5-membered ring intermediate, their energy barriers are 0.26 eV, 0.24 eV, and 0.23 eV, respectively. This indicates that the isomerization of PAMS easily occurs during degradation, especially the formation of polycyclic intermediates. In the two HAT reactions at the tail end, although the energy barriers for Htra-ta1 and Htra-ta2 are both 0.60 eV, considering that degradation is a reversible reaction, the relative energies of the product and TS (called the reverse energy barrier) also affect the occurrence of the reaction. The greater reverse energy barrier of Htra-ta2 (0.86 eV) than that of Htra-ta1 (0.67 eV) suggests that H atoms on the methylene are more likely to transfer, reflecting the selectivity of the HAT reaction. This rule also applies to Htra-he1 and Htra-he2, for which the corresponding energy barriers of these two reactions are 1.39 eV and 1.25 eV. Additionally, the HAT reaction is more likely to occur at the tail end. In a previous study,30 we have investigated the depolymerization reactions of PAMS. The energy barriers required for the dissociation of the head and tail end to generate monomer are Dhe (0.68 eV) and Dta (0.82 eV), respectively. By comparison, it was found that the tail end is more prone to occur HAT rather than depolymerization.

In fact, polymers can not only achieve HAT by overcoming the energy barrier but also instantaneous H transfer through quantum tunneling. Using different excitation energies, we calculated the probabilities of quantum tunneling (Ptunneling) for PAMS to achieve HAT and monomer dissociation based on the Wentzel–Kramers–Brillouin (WKB) approximation (Wentzel, 1926; Kramers, 1926; Brillouin, 1926). The concise tunneling model and calculation details can be seen in Figure S4 and STAR Methods. Here, the provided energy (Ep) should be higher than the energies of the reactant and product in the reaction, thus giving Ptunneling, as shown in Figure 2A. With the increase of Ep, Ptunneling increases gradually. Since only the high tunneling probability is meaningful, we will not further discuss the low tunneling probability part under the low Ep. When the same excited energy is provided, the Ptunneling of tail-end HAT (Htra-ta1 and Htra-ta2), depolymerization (Dhe and Dta), and head-end HAT (Htra-he1 and Htra-he2) decreases sequentially. The magnitudes of Ptunneling for Htra-he1 and Htra-he2 are similar. Considering that the energies of the products for these two reactions are relatively large, the required Ep is also high, thus, we will not discuss their Ptunneling in detail later. Moreover, when the Ep is 0.4 eV, the magnitude of Ptunneling for Htra-ta2 and Htra-ta1 are 10−1 and 10−4, and those for Dhe and Dta are 10−18 and 10−33, showing obvious differences. Therefore, compared with the dissociation of the monomer, the HAT reaction at the tail end has a strong tunneling effect, which can easily occur through tunneling.

Figure 2.

Figure 2

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Probabilities of quantum tunneling (Ptunneling) and thermal disturbance (Pthermal) of reaction paths

(A) Ptunneling as a function of the provided energy (Ep). “∼” denotes the magnitude.

(B) Ratio of Ptunneling to Pthermal of reaction paths when Ep is 0.40 eV. The black dotted line means that the two kinds of probabilities are equal.

Meanwhile, from the classical perspective, the thermal effect will cause the reaction to occur with a certain probability. We further compared the thermal disturbance probability (Pthermal) and Ptunneling by calculating the ratios of the two at different temperatures and Ep. Here, to clearly see the difference between Pthermal and Ptunneling, we ignore the possible correlation between the two, which has also been implemented in many researches (Guo et al., 2016; Wang et al., 2021). Figure 2B shows the ratios when the Ep is 0.40 eV. It can be found that the temperatures at which tunneling dominates in Htra-ta2, Htra-ta1, Dhe, and Dta are approximately 410 K, 340 K, 100 K, and 80 K. This suggests that H tunneling during polymer degradation can play a significant role even at room temperature. Similar laws are also obtained under other Ep (Figures S5 and S6). Actually, it is now well established that H tunneling contributions can be substantial at room temperature (McMahon, 2003; Meisner and Kästner, 2016; Schäfer et al., 2017), but this effect was not previously noticed in polymer degradation. The main role of quantum tunneling in the HAT reaction indicates that PAMS may undergo frequent HAT even at temperatures where degradation does not occur. Moreover, it is worth mentioning that the occurrence of HAT will lead to the change in the structure for the PAMS, resulting in the uncertainty of degradation. Therefore, it can be reasonably speculated that the HAT reaction promoted by quantum tunneling may be an important source of uncertainty in PAMS degradation.

In order to better understand the relevant bond properties, we summarized the bond lengths and bond orders of related C-C bonds in HAT reactions (Table S2). Here, the carbon atom on the group whose hydrogen atom is lost is named as the donor carbon atom. By taking Htra-ta1 as an example, as the H transfer reaction proceeds, the bond length of the C-C bond at the position of the donor C atom is shortened from 1.54 Å to 1.50 Å and the bond orders are enlarged from 1.00 to 1.04, while the bond lengths of its two adjacent C-C bonds are respectively increased from 1.57/1.57 Å to 1.58/1.59 Å, and the bond orders are reduced from 0.97/0.97 to 0.96/0.94. In addition, in all C-C bonds, except for the C–C bond at the monomer linkage site of the head end, these two adjacent C-C bonds are relatively weak, indicating that they are most likely to be broken. The above results indicate that after the H transfer occurs, the C-C bond where the donor C atom is located will be strengthened, and the adjacent C-C bonds will usually be weakened, leading to the occurrence of chain scission at the weaker C-C bond. The other three HAT reactions have similar bond properties (Table S2). Furthermore, we study the chain scission paths after HAT from the perspective of PESs. Also taking Htra-ta1 as an example, there are two possible scission paths after HAT. In the first pathway, a monomer free radical and a chain with a C-unsaturated end are produced. In the second pathway, a dimer and an unsaturated chain form together. The chain scission reactions that occur after the other three HAT reactions also produced various products other than monomers, such as monomer-like, dimer-like, and trimer-like molecules (Figure S7). The analyses of PESs further confirm that HAT contributes to the occurrence of chain scission reactions, thus resulting in the uncertainty of degradation.

Classical kinetic or thermodynamic control is conventionally considered an important way to understand and predict the diversity of chemical reactions. As an increasing number of tunneling phenomena in chemical reactions were revealed, researchers have gradually realized that tunneling control may be regarded as a type of nonclassical kinetic control 17,25. Moreover, an experimental study of double-hydrogen tunneling in porphycene molecules based on the scanning tunneling microscope (STM) found that the height of the STM tip affects the tunneling properties 27. Thus, it is conceivable that in future experiments, tunneling can be manipulated through external stimuli, which provides a new idea for regulating the degradation process of PAMS.

Conclusions

In summary, our study revealed the important effect of H atom tunneling on PAMS degradation. The calculation of PESs shows that there are two possible HAT paths at each end of the main chain, namely, the H atom on the methyl or methylene of ortho monomer transfers to the head end and tail end. Among them, for two reactions at the same chain end, the H atom on methylene is more likely to transfer, which reflects the selectivity of the HAT reaction. While, by comparing the energy barriers of reactions at different chain ends, it can be found that HAT occurs more easily at the tail end. More importantly, although the energy barriers required for tail-end HAT and depolymerization differ slightly, their tunneling probabilities are vastly different. At a provided energy of 0.4 eV, the tunneling probability of the former is 14–32 orders of magnitude greater than that of the latter. In particular, the tunneling effect of the former can still dominate at room temperature. Thus, it is known that lower energy barrier and stronger high-temperature tunneling effect result in a reaction in which H atom transfer to the tail end occurs more easily. Additionally, HAT will further promote the occurrence of chain scission to form products other than monomers, thus bringing about the uncertainty of degradation. The above results reveal that tunneling is an important source of the complexity of PAMS degradation, which will provide a guide to control the PAMS degradation for the better fabrication of ICF target. Meanwhile, this study will bring useful insights into solutions to the environmental pollution caused by the incomplete degradation of plastics.

Limitations of the study

This study reveals that high-temperature tunneling may be an important source of uncertainty in the degradation of polymers containing hydrogen and carbon elements. Considering the negative correlation between tunneling and atomic mass, it limits the extension of this conclusion to polymers composed of other elements to a certain extent. Hence, the effect of tunneling on the degradation of a wide range of polymer systems remains to be explored in future research.

STAR★Methods

Key resources table

REAGENT or RESOURCE SOURCE IDENTIFIER
Software and algorithms
Gaussian 09 http://gaussian.com/ N/A
DFTB+19.1 https://dftbplus.org/about-dftb/ N/A
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Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Z.G. Wang (wangzg@jlu.edu.cn).

Materials availability

This study did not generate new reagents.

Method details

DFT calculations

To carry out this study, the 3rd generation dispersion (Grimme et al., 2010) corrected density functional theory (DFT-D3) of first-principles was used to investigate the reaction paths. All structures (including reactants, transition states (TSs) and products) were fully optimized at the B3LYP-D3 (Lee et al., 1988; Becke, 1993) level in conjunction with 6-31+G(d,p) basis sets. B3LYP is a commonly used functional for studying di-radical systems, which can produce reasonable energetic results for di-radical intermediates and transition states (Beno et al., 1998; Yu et al., 2003; Siebert, 2010). Based on the optimized geometries, the nature of extreme points was evaluated by performing harmonic vibrational frequency calculations. The numbers of imaginary frequencies were used to confirm that the structure was a stable point or TS. Meanwhile, the zero-point energy (ZPE) scaled by the ZPE scaling factor was also determined. The IRC (Fukui, 1970; Gonzalez and Schlegel, 1990) was calculated to ensure the reliability of the reaction process. Due to the existence of spin contamination in the low-spin state, we corrected it with an approximate spin-projection (AP) method. AP is an effective way to eliminate spin contamination from the broken-symmetry (BS) solution (Yamaguchi et al., 1988; Takano et al., 2001). According to this scheme, the energies of the pure singlet (ES) is given below:

ES=EBS+<S2>BS<S2>T−<S2>BS(EBS−ET)

Among them, EBS and ET represent the energies of broken-symmetry state and triplet state. <S2>BS and <S2>T are spin eigenvalues of broken-symmetry state and triplet state. No atoms are frozen and no symmetry restrictions are imposed in all calculations. The above calculations were all performed using the Gaussian 09 package (Frisch et al., 2009).

Dynamic simulation

For the dynamic simulation, the DFTB+19.1 package (Hourahine et al., 2020) was used based on the empirical dispersion corrected density functional tight-binding (DFTB-D) method (Elstner et al., 1998, 2003). For systems selected in this paper, we chose the Slater-Koster type parameters (Elstner et al., 1998), which were suitable for describing organic and biological molecules. The DFTB method based on this parameter describes the geometric structure of the system containing elements such as C, H, O, and N in good agreement with the DFT method (Krüger et al., 2005). The dispersion effect was achieved by the Lennard-Jones potential (Lyuben et al., 2005) based on the UFF force field (Rappe et al., 1992). Nosé-Hoover chain thermostat (Martyna et al., 1996) and NVT ensemble were used in all kinetic calculations. The simulation step is 1 fs, the simulation time is not less than 300 ps.

Tunneling and thermal disturbance probabilities

For tunneling probability (Ptunneling), under the premise of the Born-Oppenheimer approximation, they are calculated based on WKB approximation (Wentzel, 1926; Kramers, 1926; Brillouin, 1926), as following formula (Griffiths, 2005): Ptunneling=Exp[−2ℏ∫x1x22m(V(X)−Ep)dx]. Among them, m is the reduced mass on the vibration mode corresponding to the imaginary frequency of the TS. This vibration mode contains the main degrees of freedom involved in HAT or depolymerization. Ep is the provided energy assigned to the vibrational mode along the reaction path. V(x) corresponding to the potential energy curve along the reaction path, which is obtained according to the IRC approach (Fukui, 1970; Gonzalez and Schlegel, 1990). This approach has been widely used in the analysis and prediction of a variety of chemical reactions, and shows sufficient reliability (Niu and Hall, 2000; Ziegler and Autschbach, 2005; Zhang et al., 2021; He et al., 2021). Based on the reaction paths, the expression of V(x) is obtained by the Gaussian fitting. x1 and x2 are abscissas of the two intersection points for E=Ep and E=V(x). For the thermodynamic model, thermal disturbance probability (Pthermal) suits the Boltzmann distribution, as following formula: Pthermal=Exp(−ΔE/kbT) (Landau and Lifshitz, 1980), where ΔE is the difference from Ep to the energy barrier.

Quantification and statistical analysis

Analyses and plots were performed with MATLAB and Photoshop.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant numbers 11974136 and 11674123). Z.W. also acknowledges the assistance of the High-Performance Computing Center of Jilin University and National Supercomputing Center in Shanghai.

Author contributions

Y. Zhu performed simulations and calculations; Z. Wang initiated and designed the work; Z. Wang and Z. Zhang supervised the work; Y. Zhu, X. Yang, F. Yu, R. Wang, Q. Chen, Z. Zhang, and Z. Wang analyzed the data; Y. Zhu, Z. Zhang and Z. Wang wrote the article.

Declaration of interests

The authors declare no competing interests.

Published: January 21, 2022

Footnotes

Supplemental information can be found online at https://doi.org/10.1016/j.isci.2021.103674.

Contributor Information

Zhanwen Zhang, Email: bjzzw1973@163.com.

Zhigang Wang, Email: wangzg@jlu.edu.cn.

Supplemental information

Document S1. Figures S1–S7 and Table S2
mmc1.pdf (770.2KB, pdf)
Table S1. Coordinates of the structures in reactions, related to Figure 1
mmc2.xlsx (74.3KB, xlsx)

Data and code availability

Data - Data reported in this paper will be shared by the lead contact upon request.

Code - No new code was generated during the course of this study.

Other - Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Document S1. Figures S1–S7 and Table S2
mmc1.pdf (770.2KB, pdf)
Table S1. Coordinates of the structures in reactions, related to Figure 1
mmc2.xlsx (74.3KB, xlsx)

Data Availability Statement

Data - Data reported in this paper will be shared by the lead contact upon request.

Code - No new code was generated during the course of this study.

Other - Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

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