Negative and positive anisotropic thermal expansion in 2D fullerene networks Armaan Shaikh,1, 2 Jiaqi Wu,3 and Bo Peng4, ∗ 1Homerton College, University of Cambridge, Hills Road, Cambridge, CB2 8PH, UK 2Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA 3Peterhouse, University of Cambridge, Trumpington Street, Cambridge CB2 1RD, UK 4Theory of Condensed Matter Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK (Dated: August 18, 2025) We find a design principle for tailoring thermal expansion properties in nearly-spherical molecu- lar networks. Using 2D fullerene networks as a representative system, we realize positive thermal expansion along intermolecular [2+ 2] cycloaddition bonds and negative thermal expansion along intermolecular C−C single bonds by varying the structural frameworks of molecules. The micro- scopic mechanism originates from a combination of the framework’s geometric flexibility and its transverse vibrational characteristics. Based on this insight, we find molecular networks beyond C60 with tunable thermal expansion. These findings shed light on the fundamental mechanisms governing thermal expansion in molecular networks towards rational materials design. Thermal expansion is a fundamental property of ma- terials that indicates increases in length, area, or volume upon heating, which is important in applications such as construction [1], seismographs [2], and aerospace de- sign [3, 4]. Positive thermal expansion occurs as a re- sult of an anharmonic potential energy surface, where interatomic distance increases with increasing temper- ature [5, 6]. Counterintuitively, some materials exhibit negative thermal expansion, where increasing tempera- ture leads to a contraction along certain crystallographic directions [7–15]. Such behaviors are attributed to flexi- ble crystalline networks [16, 17], rigid unit modes [18–21], and transverse displacements of bridging atoms [22, 23] or membranes [24, 25]. However, a general design principle for developing materials with negative thermal expansion is still lacking. Recent synthesis of monolayer C60 networks [26] pro- vides new avenues for designing materials with tunable thermal expansion. These networks exhibit diverse crys- talline frameworks [26–34] with nearly-spherical, stable units [35, 36] beyond rigid unit modes, as well as vari- ous intermolecular bridge bonds [37] with tunable trans- verse displacements. Compared to thermal expansion in C60 molecules and solids [38–41], the thermal behavior of C60 monolayers has yet to be understood. In 2D form, the rotational degree of freedom of C60 leads to different types of intermolecular bonds in varied crystalline net- works [Fig. 1(a)]. In this context, it would be insightful to study whether thermal expansion in molecular net- works can be controlled by intermolecular bridge bonds. Here we show that intermolecular bonds govern ther- mal expansion behaviors in fullerene networks. Inter- fullerene [2+ 2] cycloaddition bonds yield positive ther- mal expansion, whereas thermal contraction is found ∗ bp432@cam.ac.uk FIG. 1. (a) Crystal structures and (b) thermal expansion of monolayer qTP and qHP C60 networks. The schematics in (a) show the structural changes with increased temperature. along single bonds. We identify the microscopic mecha- nism by analyzing the geometric flexibility of these bridge bonds. Remarkably, we find that the low-frequency transverse vibrations along the single bonds favor ther- mal contraction, in contrast to the transverse vibrations associated with the [2+ 2] cycloaddition bonds. Based on this understanding, we rationally design molecular net- mailto:bp432@cam.ac.uk 2 FIG. 2. (a) Structural parameters of monolayer qTP fullerene networks and (b) their variations under strains. (c) Structural parameters of monolayer qHP fullerene networks and (d) their variations under strains. works beyond C60 with tailored thermal expansion. Thermal expansion is simulated under the quasi- harmonic approximation [42, 43] with volume-dependent phonons computed from density functional perturbation theory [44, 45] using VASP [46, 47]. The Gibbs free en- ergy is obtained by finding the unique minimum value of the Helmholtz free energy [48–50] with varied lattice constants a and b at a strain step of 0.2% [28, 51]. Figure 1(a) shows the crystal structures of two distinct networks of C60 monolayers, namely, the quasi-tetragonal phase (qTP) and quasi-hexagonal phase (qHP). For qTP, each carbon cage is connected by vertical and horizon- tal [2+2] cycloaddition bonds along the a1 and a2 direc- tions, respectively. Such bonds are expected to expand rigidly along their axis upon heating. In qHP, only the buckyballs along a2 are linked by the [2+2] cycloaddi- tion bonds, while C−C single bonds link the neighboring cages along a1. Figure 1(b) shows the thermal expan- sion of the two phases. While qTP along both directions has positive thermal expansion, negative thermal expan- sion along a1 is found for qHP up to 500K, in contrast to the positive thermal expansion along a2 (here we ne- glect the tiny negative thermal expansion along a2 which might come from the soft vibrational modes of the iso- lated molecule with ellipsoidal deformation that keeps the surface area constant [40, 41]). To understand the thermal expansion behaviors, we study the geometric flexibility of the two phases. Fig- 3 ure 2 shows the variations of structural parameters for qTP and qHP monolayer networks at varied strains along a1 and a2. These structural parameters measure the geo- metric flexibility of both intermolecular bonds (L1,2) and individual molecules (W1,2) along a1 and a2, as shown in Fig. 2(a) and (c) for qTP and qHP, respectively. For qTP, both the intermolecular bond L1 and the molecular width W1 expand rigidly upon uniaxial strains along a1. However, ∆L1 and ∆W1 remain nearly un- changed for strains along the other direction a2, as shown by the color map in Fig. 2(b). The same conclusion also holds for L2 and W2. For qHP, the intermolecular [2+ 2] cycloaddition bonds L2 and molecular width W2 also expand rigidly for parallel strains along a2, while resisting deformations for perpendicular strains along a1, as shown in Fig. 2(d), exhibiting behaviors similar to the [2+ 2] cycloaddition bonds in qTP. However, the intermolecular single bonds L1 and molecular widthW1 in qHP respond differently to the strains. As shown by the ∆L1 color map in Fig. 2(d), the L1 in qHP C60 becomes shorter with increased strain along a2. This notable contraction along L1 with increas- ing a2 indicates that the single bonds deform more read- ily, with compression specifically along a1 being favor- able for positive strains along a2. This hinge-like motion is expected in the single bonds as they are less resistant to perpendicular strains. On the other hand, the W1 in qHP C60 expands upon strains along a2. Therefore, we can attribute negative thermal expansion in qHP to the geometric flexibility of the intermolecular single bonds instead of the molecules themselves. The overall behavior demonstrates general features for different types of intermolecular bonds. The [2+2] cy- cloaddition bonds expand rigidly when strain is applied parallel to their direction while resisting deformations against perpendicular strains. Therefore, qTP C60 net- works exhibit nearly-isotropic positive thermal expan- sion. On the other hand, the single bonds allow for hinge- like compression when applying perpendicular strains. Unlike qTP, the flexible single bonds in qHP contract like hinges along a1 when the rigid [2+ 2] cycloaddition bonds along a2 expand upon heating. Our findings demonstrate a distinctive interplay between flexibility-driven lattice contraction via C−C single bonds and rigidity-induced structural expansion through [2+2] cycloaddition bonds. This drives a strong anisotropic thermal response in qHP C60, as summarized by the schematics in Fig. 1(a). To further explore the microscopic mechanism of neg- ative and positive anisotropic thermal expansion in qHP fullerene networks, we examine their vibrational modes. At 300 K, only phonons below 0.6 (4.3)THz have an oc- cupation number above 10 (1) according to the Bose- Einstein distribution. Thus, we focus on low-frequency phonons hereafter. Figure 3 shows the phonon dispersion curves and the corresponding mode Grüneisen parame- ters γ. The phonon dispersion along Y–Γ corresponds to FIG. 3. Low-frequency phonons, mode Grüneisen parame- ters, and vibrational modes along Γ–X. the vibrations associated with the intermolecular [2+ 2] cycloaddition bonds along a2. The mode Grüneisen pa- rameters of these vibrations are either positive or near zero, leading to thermal expansion. In contrast, the transverse displacements associated with the intermolec- ular single bonds have large negative γ along Γ–X. There are two transverse phonon branches with the largest neg- ative γ: an out-of-plane acoustic mode (ZA) with the co- herent movement of all molecules along z, and an out- of-plane optical mode (ZO) with alternating displace- ments between neighboring molecules, as illustrated by the schematics in Fig. 3. Both modes with large negative γ favor lattice contraction. In comparison, the longitudi- nal acoustic mode (LA) has positive γ, which contributes to lattice expansion instead but less strongly (γ < 1.5) than the ZA and ZO modes (|γ| > 2.5). The overall pic- ture confirms that transverse displacements of the sin- gle bonds yield thermal contraction, while the [2+ 2] cy- cloaddition bond oscillations contribute to expansion be- haviors. The rigidity of the [2+ 2] cycloaddition bonds, as well as the flexibility of the C−C single bonds, pro- vide a universal design principle to tailor thermal ex- pansion behaviors. The [2+ 2] cycloaddition bonds be- tween molecular cages impose structural, elastic, and vi- brational constraints that only allow positive thermal ex- pansion. Contrastingly, the flexible single bonds bridging the molecules allow for lattice contraction when perpen- dicular strains are applied, and the vibrational modes as- sociated with these single bonds yield strong transverse displacements that favor thermal contraction. The flex- ibility of the single bonds is therefore the main driving factor in negative thermal expansion. Similar thermal be- haviors in fullerene-based networks have also been found in previous molecular dynamics simulations [52]. Intu- itively, we can either realize positive thermal expansion through intermolecular [2+ 2] cycloaddition bonds to re- sist lattice contraction, or utilize less rigid intermolecular 4 FIG. 4. (a) Crystal structures and (b) thermal expansion of 2D qHP monolayer C24 and C60 networks. single bonds to yield negative thermal expansion. The discovery that C−C single and [2+2] cycloaddi- tion bonds contribute distinctively to thermal behavior provides a predictive tool for the rational design of ther- mally responsive materials with nearly-spherical build- ing blocks such as icosahedral B12 units [53–55], fullerene cages [56, 57], and all-metal clusters [58] (for thermal ex- pansion of 2D icosahedral B12 networks, see Fig. 5 in the End Matter). As a proof-of-principle study, we extend this design principle to monolayer qHP C24 networks, where the C24 molecules are linked through similar single bonds along a1 but three intermolecular bonds between six carbon atoms along a2 [59]. The crystal structure of qHP C24 is shown in Fig. 4(a). Unlike the nearly-planar single bonds in qHP C60, the single bonds in C24 mono- layers have a buckled structure owing to a larger molecu- lar curvature and asymmetric intermolecular bonding po- sitions. This, in combination with the smaller molecular size, leads to a higher density of interfullerene bonds and larger elastic constants than qHP C60 [59]. It is therefore expected that the single bonds in C24 are more rigid than those in qHP C60, leading to stronger resistance to per- pendicular deformations. Figure 4(b) shows smaller neg- ative thermal expansion in C24 compared to C60 along a1 as expected. Additionally, the positive thermal ex- pansion along a2 in C24 is also larger than that in C60. This is unsurprising since the three intermolecular bonds along a2 in C24 are much more rigid than the [2+ 2] cy- cloaddition bonds in monolayer qHP C60 networks. The recent experimental realization of monolayer fullerene networks [26] offers a timely opportunity to di- rectly test our predictions, which influence numerous fullerene-based applications such as device integration in photodetectors [60] and catalytic activity in photocatal- ysis [27, 29, 30, 61]. The temperature-dependent struc- tural properties can be measured using high-resolution capacitance dilatometers [38] and synchrotron X-ray diffraction [39]. The low-frequency phonon modes can be detected by Raman and infrared spectroscopy [41], en- abling verification of the predicted transverse vibrational mechanisms. From an application perspective, molecular networks with tunable thermal expansion present com- pelling possibilities. Materials with designed negative or near-zero thermal expansion are in high demand in flexi- ble electronics [2], aerospace composites [3], and precision instrumentation [62], where thermal mismatch must be minimized. Our results indicate that such properties can be achieved not through complex multi-phase compos- ites, but through intrinsic molecular architecture. Previ- ous experimental data on thermal expansion in molecu- lar networks [7, 10, 12, 16, 17, 63–66] including polymeric fullerene [38, 67] confirm the fact that thermal expansion in molecular crystals can be controlled by bonding motifs and crystalline geometries, and similar mechanisms have been observed experimentally such as flexibility-driven “hinge-like” motion in crystalline networks [7, 10, 16, 17] and transverse displacements of bridging atoms [22, 23] or membranes [24, 25]. This, again, supports the broader generalisability of our proposed mechanism. In summary, we establish a general method for en- gineering thermal expansion in cage-like molecular net- works through different intermolecular bonds. Using fullerene as a representative system, we show that in- termolecular [2+ 2] cycloaddition bonds result in posi- tive thermal expansion, whereas C−C single bonds lead to negative thermal expansion. By varying the struc- tural frameworks of fullerene molecules, we can de- sign a nearly-square lattice with positive thermal expan- sion through [2+2] cycloaddition bonds along both in- plane directions. Similarly, we can also realize negative and positive anisotropic thermal expansion in a nearly- triangular lattice through single bonds along one direc- tion and [2+2] cycloaddition bonds along the other di- rection. The origin is related to both the geometric flex- ibility of different intermolecular bonds and their corre- sponding transverse vibrations. By identifying and un- covering this microscopic mechanism, we can further de- sign molecular networks with tailored thermal expansion. A.S. acknowledges support from Homerton College Cambridge for a Homerton-Victoria Brahm Schild Grant. J.W. acknowledges support from the Cambridge Under- graduate Research Opportunities Programme and from Peterhouse for the James Porter Scholarship. B.P. ac- 5 knowledges support from Magdalene College Cambridge for a Nevile Research Fellowship. 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We explic- itly compute thermal expansion of monolayer qHP B12 networks and find the same trend in Fig. 5(b). FIG. 5. (a) Slightly rotated top view of the crystal struc- tures (to show the double intercluster bonds) and (b) thermal expansion of monolayer icosahedral B12 networks. Negative and positive anisotropic thermal expansion in 2D fullerene networks Abstract Acknowledgments References Thermal expansion of 2D icosahedral B12 networks