Structure and Dynamics of a Graphene Melt

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Structure and Dynamics of a Graphene Melt Wenjie Xia, Fernando Vargas Lara, Sinan Keten, and Jack F. Douglas ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.8b00524 • Publication Date (Web): 22 May 2018 Downloaded from http://pubs.acs.org on May 22, 2018

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Structure and Dynamics of a Graphene Melt Wenjie Xia,1,2,3* Fernando Vargas-Lara,1 Sinan Keten,3,4* Jack F. Douglas1* 1

Materials Science & Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899

2

Center for Hierarchical Materials Design, Northwestern University, Evanston, IL 60208-3109

3

Dept. of Civil & Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3109

4

Dept. of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3109 *To whom correspondence should be addressed. Emails: [email protected], [email protected], [email protected]

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ABSTRACT We explore the structural and dynamic properties of bulk materials composed of graphene nanosheets using coarse-grained molecular dynamics simulations. Remarkably, our results show clear evidence that the bulk graphene materials exhibit fluid-like behaviors similar to linear polymer melts at elevated temperatures and that these materials transform into a glassy-like “foam” state at temperatures below the glass-transition temperature (𝑇𝑔 ) of these materials. Distinct from an isolated graphene sheet, which exhibits a relatively flat shape with fluctuations, we find that graphene sheets in a melt state structurally adopt more “crumpled” configurations and correspondingly smaller sizes, as normally found for ordinary polymers in the melt. Upon approaching the glass transition, these two-dimensional “polymeric” materials exhibit a slowing down of their dynamics that is likewise similar to ordinary linear polymer glass-forming liquids. Bulk graphene materials in their glassy “foam” state have an exceptionally large free-volume and high thermal stability due to their high 𝑇𝑔 ( ≈ 1 600 K) comparing to conventional polymer materials. Our findings show that graphene melts have interesting lubricating and “plastic” flow properties at elevated temperatures, and suggest that graphene foams appear to be highly promising as high surface filtration materials and fire suppression additives for improving the thermal conductivities and mechanical reinforcement of polymer materials.

Keywords: graphene sheet, foam, glass transition, structure, coarse-grain, molecular dynamics simulation


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Recently, there has been a particular upsurge of interest in graphene sheets, which have many excellent thermal, mechanical and electrical properties for advanced applications in electronics,1 energy storage,2 impact protections,3 and nanocomposites.4 Graphene sheets can be obtained in large quantities from the liquid-phased exfoliation of common graphite and be separated by size at a relatively low cost and environmental impact in comparison to other carbon materials (e.g., carbon nanotube). At the same time, numerous other nanosheet materials (e.g., graphene-oxide, molybdenum disulfide (MoS2), boron nitride (hBN), etc.)5 have now been synthesized based on exfoliation,6 which have a wide range of properties, and graphene is a particular representative of a whole family of nanosheet materials. While molecular cartoons often depict graphene as a rigid crystalline two-dimensional sheet, this picture can be misleading for understanding the structural and dynamic properties of these materials. While this “macromolecule” has a remarkably high inplane mechanical stiffness (i.e., a Young’s modulus of about 1 TPa),7 its out-of-plane bending rigidity (about 1.5 eV) is relatively low, comparable to that of lipid bilayers in cells, due to its atomically thin nature.8 In reality, we should view graphene sheets as “two-dimensional polymers”9-13 (synthesized through surface-assisted polymerization of monomers), where conformations become “crumpled” and “disordered” due to the thermal energy, external stress or interactions with a surrounding solvent or material matrices in composites.14-19 This is also true for graphene-oxide20 and other nanosheet materials (e.g., MoS2)19 which are even more flexible than graphene. Such crumpling and wrinkling phenomena are also widely observed in everyday objects, such as paper,21 thin polymer films,22, 23 and biological and polymerized membranes.24, 25 Most of recent efforts have been focused on investigating graphene-based bulk materials near room temperature (e.g., graphene “aerogels”,26 “inks”27 and “foams”28 – formed from the dispersal of graphene sheets in a carrier fluid). However, their behavior at elevated temperatures is also of


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great interest given the applications of graphene and other nanosheet materials operating under high temperature (well above room temperature) conditions where the material is in its fluid state (we determine this condition below for a graphene melt). This high temperature regime is particularly relevant for energy conversion and neutron moderator materials for nuclear reactor,29 flammability suppressing additives for polymer materials,30 gas filtration and isotope separation media,31 etc. Since graphene sheet can be considered as a two-dimensional polymer, it seems natural to consider whether these bulk materials exhibit fluid-like properties similar to ordinary polymers having a linear chain architecture. As in the case of polymers, we may expect the crumpling and disordering to become enhanced in form of a “melt” of graphene sheets at elevated temperatures. The present work aims to address the structural and dynamical nature of graphene sheets in its liquid-like melt form at elevated temperatures. In particular, we seek to answer whether such graphene “melts” exhibit characteristic glass-forming properties analogous to polymers.32 Recent studies have shown some evidence that bulk graphene materials exhibit rubberlike elasticity and rheological response similar to polymers.33 Understanding the structures and dynamics of graphene sheets in solution and in the melt is crucial to develop an extension of structure-property relationships as established for linear polymers.


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Figure 1. (a) Mapping from the all-atomistic model (grey atoms) to the CG model (blue beads) of graphene. Four connected carbon atoms (highlighted on the right) are grouped into a CG bead. (b) Initial configuration of the single CG graphene nanosheet that has a length of 48 nm and a width of 8 nm. (c) Snapshot of a graphene melt showing the disoriented and crumpled sheets. Each sheet is colored differently to aid the visualization.

In the present work, we perform molecular dynamics (MD) simulations to investigate the graphene melt by employing our previously developed coarse-grained (CG) model of graphene.34 The CG model has been derived based on a 4-to-1 mapping scheme from an atomistic graphene model by conserving the elastic strain energy (i.e., the so-called strain energy conservation


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approach), which preserves the hexagonal lattice geometry of graphene (Figure 1a). The functional forms and parameters of the CG model, including bonds 𝑉𝑏 , angles 𝑉𝑎 , dihedrals 𝑉𝑑 , and nonbonded interactions 𝑉𝑛𝑏 , are listed in Table I. This CG model accurately reproduces the elastic, fracture, interlayer shear, and adhesion properties of graphene sheets, and the model predictions are broadly in good agreement of experiments and other simulation results.34-37 Our CG model of graphene achieves about hundreds of fold increase in computational speed in comparison to the all-atomistic simulations, thus allowing for access to greater spatiotemporal scales. Based on this CG model, we systematically investigate the temperature ( 𝑇 ) dependent thermomechanical behaviors of bulk graphene material, in which graphene nanosheets having a dimension of about 8 nm in width and 48 nm in length (Figure 1b) are packed into randomly oriented configurations forming the graphene “melt” (Figure 1c). (See more description of the model and simulations in the Methods section.) In particular, we examine the dynamics of the graphene melt spanning over a wide temperature range from the high-𝑇 regime, where the sheets behave as a fluid, to the low𝑇 regime, where the sheets approach the glassy state. Our simulations show the 𝑇-dependence of dynamics and mechanical responses of graphene melts are qualitatively similar to ordinary linear polymers. Remarkably, we find clear evidence that graphene melt exhibits the glass-forming characteristic dynamics of a liquid over a large 𝑇 range above the glass-transition temperature (𝑇𝑔 ). Below the 𝑇𝑔 , the graphene melt is transformed into a “foam” at a glassy state. Our simulation demonstrates an exceptional high 𝑇𝑔 and thus thermal stability of graphene foam comparing to ordinary polymers, which has great potential applications at extreme thermal conditions.

Table I. Functional forms and parameters of the coarse-grained graphene model.


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Interaction

Functional Form

Parameters 𝑑0 = 2.8 Å

Bond

𝑉𝑏 (𝑑) = 𝐷0 {1 − exp⁡[−𝑎(𝑑 − 𝑑0 )]}2

𝐷0 =196.38 kcal/mol

for 𝑑 < 𝑑𝑐𝑢𝑡

𝑎 = 1.55 Å-1

𝑑𝑐𝑢𝑡 = 3.49 Å 𝑉𝑎 (𝜃) = 𝑘𝜃 (𝜃 − 𝜃0 )2

Angle

𝜃0 = 120° 𝑘𝜃 = 409.4 kcal/mol

Dihedral

𝑉𝑑 (𝜑) = 𝑘𝜑 [1 − cos⁡(2𝜑)]

𝑘𝜑 = 4.15 kcal/mol 𝜎 = 3.46 Å

Non-bonded

𝜎 12 𝜎 6 𝑉𝑛𝑏 (𝑟) = 4𝜀 [( ) − ( ) ] 𝑟 𝑟

𝜀 = 0.82 kcal/mol

for 𝑟 < 𝑟𝑐𝑢𝑡

𝑟𝑐𝑢𝑡 = 12 Å

RESULTS AND DISCUSSION Morphology of an Isolated Graphene Sheet vs. Melt. In this section, we first explore the morphological characterizations of an isolated graphene sheet in vacuum at elevated temperature (𝑇 ≈ 3 000 K) to model the sheet in a gas phase or dispersed in a solution.27 In particular, we focus our attention on basic flow properties that are relevant to the structure of the sheet, including the hydrodynamic radius 𝑅ℎ , the radius of gyration 𝑅𝑔 and intrinsic viscosity [𝜂], which are important characterization observables that can be experimentally measured using dynamic and static scattering techniques and viscosity measurements. We then investigate the structural properties of graphene sheets in the melt to better understand their morphological difference from the isolated one.


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Figure 2. (a) Hydrodynamics radius 𝑅ℎ , (b) radius of gyration 𝑅𝑔 and (c) intrinsic viscosity [𝜂] of an isolated graphene sheet at an elevated temperature 𝑇 = 3 000 K. (d), (e) and (f) are the histograms of the normalized number frequency obtained from (a), (b) and (c), respectively. The 𝑅ℎ and [𝜂] data exhibit a Gaussian distribution, and the 𝑅𝑔 data exhibits a skew normal distribution. At elevated temperature, the thermal fluctuations cause the isolated graphene sheet to wrinkle, an effect similar to the random coils of linear polymer chains. To observe the wrinkling effect in our simulations, we generate 1 600 different configurations spanning over 2 ns period via MD after equilibrating the sheet, from which we are able to calculate the basic solution characteristic properties, 𝑅ℎ , 𝑅𝑔 , and [𝜂]. Figure 2a-c show the fluctuations of these properties (𝑅ℎ , 𝑅𝑔 and [𝜂]) as a function of number of configurations. Figure 2d-f show the number frequency of these properties obtained from Figure 2a-c, respectively. The 𝑅ℎ and [𝜂] data for the sheet exhibit a Gaussian distribution, whereas the 𝑅𝑔 data shows a skew normal distribution, which can be


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attributed to the difference in the dimensions of the length and width of the sheet adopted in our study. These analyses indicate that while the graphene sheet fluctuates greatly due to the thermal effect, it maintains relatively flat rather than a collapsed conformation at elevated temperatures. The distributions of their structural and dynamic properties should be considered as important characteristics of isolated graphene sheets in solution. As a comparison, for an ideal (i.e., perfectly flat) graphene sheet adopted in our study (Figure 1b), the mean values of 𝑅𝑔 , 𝑅ℎ and [𝜂] are larger than those of an isolated sheet (Table II), indicating a more extended shape. We also simulate a fully collapsed graphene sheet (as in a “poor” solvent) by applying an effective potential that describes the many-body self-interactions of the sheet, which leads to the effective one-body potential having the form of the average density of the fractal random sheet structure whose fractal dimension in the present case is near 2.38 This corresponds to an attractive “1/𝑅-type” potential, centered at the center of mass (COM) of the sheet where 𝑅 is the distance between each CG bead and the COM of the sheet. Based on this idealized model of a graphene sheet in a poor solvent, we observe that the mean values of 𝑅𝑔 , 𝑅ℎ and [𝜂] of the collapsed sheet (Table II) are significantly smaller than those of the ideal and isolated sheets, which provide rough lower limits of these measures of nanosheet size. Of course, these solution property estimates of graphene sheet size also depends on the sheet mass, as in the case of ordinary polymers. The selection of the ribbon-like graphene sheet geometry and particularly size serves only as a representative example, motivated by computational expediency, and by the fact that graphene sheets having a similar size and shape have been experimentally synthesized before.39 We next consider a condensed fluid comprised of disoriented graphene sheets, forming the graphene melts.


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Table II. Summary of basic conformational properties of an ideal (i.e., perfectly flat), isolated and collapsed graphene sheet, and sheets in the graphene melt. The uncertainty is estimated by the standard deviation of the properties indicated. Sheet Ideal Isolated Collapsed Melt

〈𝑹𝒈 〉 (nm) 14.0

〈𝜿𝟐 〉 0.913

〈𝑹𝒉 〉 (nm) 8.8

[𝜼] 131.2

13.7±0.1

0.894 ± 0.015

8.7 ± 0.2

125.6 ± 17.8

2.3

0.004

3.2

4.7

10.7 ± 1.6

0.582 ± 0.201

-

-

Figure 3. Representative configurations of (a) a crumpled and (b) a less crumpled graphene sheet in the melt, and representative configurations of (c) an isolated and (d) a fully collapsed graphene


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sheet. Comparison of the histograms of (e) 𝑅𝑔 and (f) relative shape anisotropy 𝜅 2 for the graphene sheets in the melt and an isolated graphene sheet, respectively. For the graphene melt, the sheets exhibit more complex morphologies with disordered orientations (e.g., folded, crumpled and coiled conformations as illustrated in Figure 1c) comparing to the isolated graphene sheet discussed above. Figure 3a and 3b show the representative configurations of graphene sheets in the melt (i.e., crumpled and less crumpled) in comparison to a relatively flat isolated sheet (Figure 3c) and a fully collapsed sheet (Figure 3d) as described above. To quantify the shape of graphene sheets at melt conditions, we calculate the radius of gyration tensor 𝑹𝟐𝒈 generated from our simulations, from which we are able to evaluate not only the 𝑅𝑔 but also a commonly used shape descriptor 𝜅 2 , called the relative shape anisotropy, by calculating the eigenvalues of 𝑹𝟐𝒈 . In particular, 𝜅 2 = 0 for iso tropic spherical particles and is 𝜅 2 = 1 for linear particles. Figure 3e and 3f show the frequency distributions of 𝑅𝑔 and 𝜅 2 for the graphene melt from our simulations, respectively. We then see that the 𝑅𝑔 of graphene sheets in the melt exhibits a rather broad and non-Gaussian distribution with a mean 𝑅𝑔 value of about 10.7 nm, which lies between the sizes of the flat isolated sheet (i.e., in a good solvent) and the collapsed sheet (i.e., in a poor solvent), due to partial screening of excluded volume interactions by surrounding sheets in the melt. This observation is qualitatively analogous to the solution behavior of linear polymers, where polymer chain dimensions in the melt are intermediate between those in a good solvent where the chain is swollen and in a poor solvent where it is collapsed, due to screening of excluded volume interactions by surrounding polymers.40 Similarly, 𝜅 2 of graphene sheets in the melt has a wider distribution with an average value of around 0.6, which lies between that of an isolated and a


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collapsed sheet. The mean values of these structural properties of an ideal, an isolate and a collapsed sheet and the sheets in the graphene melt are summarized in Table II. We also performed additional simulations with much larger systems (i.e., 400 and 2 500 sheets with a similar overall density) (see Figure S1 in Supporting Information) to examine for possible finite size effects; 𝑅𝑔 and 𝜅 2 , and the uncertainty determined by the variance of these properties, are not greatly affected by the system size (𝑅𝑔 variance ≲ 1.9 nm and 𝜅 2 variance ≲ 0.2). The relatively large uncertainty estimate reflects wide distributions of the sheet conformation and sizes in the melt. This confirms that graphene sheets in the melt adopt configurational structures intermediate between an isolated “swollen” graphene sheet and a “collapsed” sheet.

Dynamics of a Graphene Melt. As noted before, the “crumpled” morphology of graphene sheets in the melt as described above is reminiscent of the “random coil” conformations of linear polymers in the “melts”. These polymers normally exhibit glass formation associated with this conformational complexity (i.e., their amorphous and disorder characteristics) and a correspondingly strong temperature dependence of their dynamics above the glass-transition temperature (𝑇𝑔 ). We next seek to answer the question of whether the graphene melt will also exhibit fluid-like dynamics similar to polymer glass-forming liquids. To do so, we evaluate the potential energy of the system by calculating the pair-wised nonbonded interactions as a function of temperature. Figure 4 shows that the time-averaged potential energy decreases with lowering the temperature. The inset in Figure 4 shows the time independency of the potential energy at different temperatures. Evidently, the bulk graphene material undergoes a glass-transition similar to that of linear polymer melts and other glass-forming liquids upon cooling, which are attributed to the conformational complexity and “packing frustration” of these macromolecules. The energy


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vs. 𝑇 can be fitted with two linear lines with different slopes in the high-𝑇 and low-𝑇 ranges, where the intersection of the two lines gives an estimate of 𝑇𝑔 (also called “thermodynamic” 𝑇𝑔 ) as shown in Figure 4. The 𝑇𝑔 of the graphene melt is estimated to be about 1 610 K (with a standard deviation of 82 K) from this energy calculation, which is much higher than most polymer materials. This finding suggests an exceptionally high thermal stability of the graphene melt under condition where oxygen is not present to oxidize the graphene.

Figure 4. Potential energy of the graphene melt as a function of temperature 𝑇. The energy values in the high-⁡𝑇 and low-⁡𝑇 regimes are fitted with two linear lines, where the intersection marks the estimation of 𝑇𝑔 . (Inset) Potential energy is nearly time independent at varying 𝑇 after equilibration of the system.

We proceed to examine the temperature-dependent dynamics of the graphene melt. It is widely appreciated that the dynamics of polymeric fluids is rather complex and exhibits strongly nonArrhenius relaxation behaviors as the temperature is lowered. Here, we evaluate the 𝑇-dependent relaxation by calculating the self-part of the intermediate scattering function 𝐹𝑠 (𝑞, 𝑡),


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1 Fs (q, t ) = N

N

 j

exp  −iq  ( r j (t ) − r j (0) ) 

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(1)

where 𝑁 is the total number of beads, 𝑟𝑗 (𝑡) is the position of the 𝑗th bead at time 𝑡, and 〈… 〉 is the ensemble average. 𝑞 = |𝒒| is the magnitude of the wavenumber taken from the first peak of the structure factor 𝑆(𝑞), which is estimated to be 𝑞 = 15.1 nm-1. The structural relaxation time 𝜏𝛼 is defined as the time where 𝐹𝑠 (𝑞, 𝑡) decays to 0.2, which is consistent with previous studies.41-43 Figure 5a shows the result of 𝜏𝛼 over a temperature range from 2 500 K to 6 000 K. As 𝑇 decreases, the relaxation dynamics becomes slower for the graphene melt, leading to dramatic increases in 𝜏𝛼 . Similar to linear polymer melts and other glass-forming liquids, remarkably, we observe that the 𝑇-dependent 𝜏𝛼 of graphene melt can be captured by the well-established Vogel-FulcherTammann (VFT) relation:44-46

 DT0    T − T0 

  (T ) =   exp 

(2)

where 𝜏∞ , 𝐷 and 𝑇0 are fitting parameters associated with glass-forming process. In particular, 𝐷 is inversely related to the fragility parameter 𝐾⁡(= 1⁄𝐷 ), a property that defines the strength of the temperature dependence of 𝜏𝛼 and its deviation from the Arrhenius relaxation. 𝑇0 , also called Vogel-Fulcher temperature, indicates the “end” of glass-formation where 𝜏𝛼 becomes extremely large. While 𝐾 is estimated to be around 0.15 that is comparable to ordinary linear polymers, 𝑇0 is about 1 200 K for the simulated graphene melt, which is much higher comparing to polymers. From the VFT relation, we can estimate 𝑇𝑔 by extrapolating the relaxation data to the empirical observation timescale, 𝜏𝛼 (𝑇𝑔 ) ≈100 s, where we find 𝑇𝑔 to be around 1 580 K (with a standard deviation of 88 K) for the graphene melt. This value is much larger than the 𝑇𝑔 ’s of commonly


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applied linear polymers, such as polystyrene. While this 𝑇𝑔 value estimated from the 𝜏𝛼 is lower than that from the potential energy calculations, considering the numerical uncertainty, both calculations yield a consistent 𝑇𝑔 (around 1 600 K). This analysis of relaxation dynamics of the graphene melt, in conjunction with the energy calculations, confirms the existence of glass formation in the graphene melt upon cooling. We also performed simulations using a larger melt system that consists of 100 sheets, and the result confirms that there is no significant system size effect on our dynamics analysis (Figure S2 in Supporting Information). The melting temperature of a graphene sheet has been reported to be about 5 000 K,47 which is much larger than the 𝑇𝑔 of the melt. Our result indicates that the bulk graphene materials have great potential for high temperature applications due to their excellent thermal stability and high 𝑇𝑔 .


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Figure 5. (a) Structural relaxation time 𝜏𝛼 of the graphene melt as a function of temperature 𝑇, which is evaluated from the self-part of intermediate scattering function 𝐹𝑠 (𝑞, 𝑡). The data is fitted with the VFT relation (Eq. 2). (Inset) The Debye-Waller factor 〈𝑢2 〉 as a function of 𝑇. (b) Test of the localization model of relaxation (Eq. 3) for the graphene melt.

It has been established that the dynamics of the glass-forming materials are strongly associated with the localized “caging” effects imposed by the neighbor segments. This localization effect can be assessed through a consideration of a fast dynamics property, the Debye-Waller Factor 〈𝑢2 〉, which quantifies the local “free volume” and molecular “stiffness” of the material at picosecond


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timescale over which the molecules are caged by their neighbors. Experimentally, 〈𝑢2 〉 can be measured via neutron scattering technique. In our simulations, we evaluate the 〈𝑢2 〉 from the mean-squared displacement 〈𝑟 2 (𝑡)〉 of the CG beads, where 〈𝑢2 〉 is defined as the magnitude of 〈𝑟 2 (𝑡)〉 at a time 𝑡 ≈ 40 ps, corresponding to the localization timescale. The inset in Figure 5a shows the temperature dependence of 〈𝑢2 〉 for the graphene melt. We observe that 〈𝑢2 〉 decreases with reducing the temperature in a nonlinear fashion, indicating a lower mobility upon cooling. This behavior is characteristically analogous to the linear chain polymer melts. Historically, it has been argued by Hall and Wolynes48 that the structural relaxation time 𝜏𝛼 of a glass-forming liquid should obey a scaling relation with 〈𝑢2 〉 via 𝜏𝛼 ⁡~⁡exp[𝑢02 /〈𝑢2 〉] with 𝑢02 as an adjustable constant. Recently, Simmons and coworkers49 argued that 𝜏𝛼 could be described by a localization model (LM) (i.e., an extension of the Hall-Wolynes model), 𝜏𝛼 ⁡~⁡exp[(𝑢02 /〈𝑢2 〉)𝛼⁄2 ], where the exponent 𝛼 is related to the shape of the free volume (e.g., 𝛼 ≈ 3 for a isotropic spherical shape). More recently, Betancourt et al.50 made a step forward by reducing 〈𝑢2 〉⁡by its value at the onset temperature 𝑇𝐴 for molecular caging and by fixing the prefactor in the 𝜏𝛼 - 〈𝑢2 〉 relation by the observed 𝜏𝛼 value⁡𝜏𝐴 at 𝑇𝐴 . Since it is difficult to quantify the 𝑇𝐴 (which is expected to be a much higher temperature) from our simulations, here we reduce 𝜏𝛼 and 〈𝑢2 〉 by their values at high temperature 𝑇 ≈ 6 000 K (the highest 𝑇 tested in our simulations) in order to test whether the prediction of LM will hold for the graphene material. This leads to a predictive relation,

(

  (T ) =  r exp  ur2 / u 2 (T )

)

 /2

− 1 

where 𝑢𝑟2 and 𝜏𝑟 are the reference values of 〈𝑢2 〉 and 𝜏𝛼 at 𝑇 ≈ 6 000 K, respectively.


(3)

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Figure 5b shows a comparison of Eq. 3 to our simulation data of relaxation, where we find reasonable agreement with the LM. This analysis is particularly useful as it provides quantitative relationship between the long-time relaxation dynamics 𝜏𝛼 and the fast dynamics property 〈𝑢2 〉 at picosecond timescales. The exponent 𝛼 in the LM is determined to be about 2.2 from the best fit of the data for the graphene melt. This exponent estimate is consistent with the Hall-Wolynes model prediction of 𝛼 ≈ 2 within numerical uncertainty, but the exact 𝛼 value should vary in general with the fluid type, which depends on the shape of free volume. A value of 𝛼 near 2 is reasonable according to the LM due to the 2D nature of the graphene sheets, since the volume in which the segment “rattles” can be expected to be sheet-like. This analysis indicates that the relaxation dynamics of the simulated graphene melt over a wide 𝑇 range can be well described by the LM of glass formation, which can also predict the anisotropic geometry of the segmental free volume for the graphene melt. While we focused on understanding the structural and dynamic properties of a graphene melt in the current study, it would be interesting to examine whether other sheet materials, such as MoS2 and hBN, exhibit fluid-like behaviors at elevated temperatures in future work.

Mechanical properties of a graphene “foam”. As in the case of polymers, at low temperatures below 𝑇𝑔 , the materials often transform into a glassy state as manifested by the enhanced mechanical properties, such as modulus. We next examine the shear properties of the graphene melt in their glassy “foam” state below the 𝑇𝑔 by shearing the system with a strain rate of 0.5 ns-1. Figure 6a shows the snapshots of the shear simulation at 0 and 0.5 shear strain. We can see that as the shear strain increases, the graphene sheets are more aligned towards the diagonal


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direction with an angle of 45° relative to the shear direction at 0.5 shear strain. From the shear testing, the shear stress vs. strain curve can be obtained as shown in Figure 6b, where the dots show the original stress output from our simulations and the curve shows the smoothed result. At a strain less than 1%, we observe that the stress increases linearly with strain, where the shear modulus 𝐺 is determined by linearly fitting the stress-strain data. As the strain increases beyond the linear elastic regime, the foam starts to yield with a yielding stress around 20 MPa at room temperature (𝑇 ≈ 300 K), followed by a strain hardening behavior. The strain hardening of the graphene foam is mainly attributed to the rearrangement and alignment of the sheets under the shear. This overall shear behavior of graphene foam is qualitatively similar to that of glassy polymers having a linear topology.


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Figure 6. (a) Snapshots of shearing of the graphene foam at 0 and 0.5 shear strain (each sheet is colored differently for visualization). (b) Shear stress vs. strain at 𝑇 = 300 K. The fluctuated grey line and dark blue curve are the original stress values and the smoothed stress result, respectively. The shear modulus 𝐺 is determined by linearly fitting the stress-strain data within 1% strain (the dashed slop). (Inset) 𝐺 as function of 𝑇 in the glassy regime (below 𝑇𝑔 ). (c) The nonlinear scaling relationship between 𝐺 and 1/〈𝑢2 〉 in the glassy regime below 𝑇𝑔 , which can be described by a power-law function (red curve).


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The inset in Figure 6b shows the shear modulus 𝐺 as a function of temperature. It can be observed that 𝐺 decreases from around 480 MPa to 250 MPa as 𝑇 increases from 100 K to 1 500 K in a nonlinear relationship. This strong temperature dependence of 𝐺 of the graphene foam is in contrast to the in-plane shear properties of an individual graphene sheet, which are found to be weakly dependent of 𝑇 in the range from 100 K to 2 000 K.51 The values of 𝐺 for the graphene foam is about three orders of magnitude lower than the in-plane shear modulus of the graphene sheet. At room temperature, the obtained value of 𝐺 ≈ 425 MPa is comparable but lower than that of glassy polymers, e.g., polystyrene and poly(methyl methacrylate) with a shear modulus of around 1 000 MPa. One reason for this observation is that the graphene foam has a much lower density (around 0.7 g/cm3) compared to the pristine graphene (around 2.2 g/cm3) and typical linear chain glassy polymers (usually above 1 g/cm3). At small strains, the shear response is mainly governed by the inter-sheet sliding and cohesive interaction between the sheets rather than stretching of the sheet, leading to the lower 𝐺 value. Experimentally, it is commonly observed that the interlayer shear modulus of stacked graphene sheets can range from around 5 MPa to 2 000 MPa depending on the stacking orientations (i.e., commensurate stacking with a higher stiffness and non-commensurate stacking with a lower stiffness). The observed 𝐺 of graphene foam is expected to lie within this range. Nevertheless, higher stiffness and fracture energy of the graphene foam can be achieved by chemically crosslinking the sheets to enhance stress transfer upon loading as reported in recent studies,52, 53 which is similar to cross-linked polymers. For the linear glassy polymers and other glass-forming liquids, previous studies have indicated a correlation between the molecular stiffness, which is inversely related to 〈𝑢2 〉, and the shear modulus of the materials. In particular, recent work on metallic glasses with different composition reported that 𝐺 is linearly scaled with 1⁄〈𝑢2 〉 at a glassy state.54 Figure 6c shows the test of the


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correlation between the 𝐺 and 1⁄〈𝑢2 〉 for the graphene foam. While we see that the 𝐺 increases with 1⁄〈𝑢2 〉, the commonly observed linear scaling relationship doesn’t hold for the graphene foam. Instead, we observe an apparent power-law relationship between 𝐺 and 1⁄〈𝑢2 〉 . This nonlinear scaling relationship for the graphene foam might be attributed to the high porosity of the system, wherein the average and maximum pore diameters are observed to be about 5 nm and 7 nm at 𝑇 = 300 K (Figure S3 in Supporting Information), respectively. This high porosity of the foam may cause that the temperature dependent local free volume and molecular caging are somehow different from the glassy polymer and metallic glasses.

CONCLUSION In the present study, we have systematically investigated the thermomechanical behaviors of the graphene melt composed of disoriented graphene nanosheets by employing the CG-MD simulations. Remarkably, our simulation results provide clear evidence that the graphene melt exhibits fluid-like properties analogous to linear chain polymers. Specifically, the graphene melt exhibits the dramatically slowing down of their relaxation dynamics upon cooling due to the glass transition, phenomena often observed in ordinary glass-forming liquids. The graphene sheets in the melt exhibit highly disoriented and crumpled conformations with large free-volume, in contrast to an isolated graphene sheet that has a fluctuated flat geometry. At lower temperatures below the glass-transition temperature, the graphene melt is transformed into a glassy “foam” state achieving a higher shear modulus. Our results demonstrate the exceptionally high glass-transition temperature (i.e., 𝑇𝑔 ≈ 1600 K) and thermal stability of the graphene foam, suggesting that this bulk graphene material can be a promising candidate for applications at extreme thermal conditions.


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METHODS Overview of the Coarse-Grained Graphene Model. Our coarse-grained (CG) model is based on a 4-to-1 mapping scheme derived from the all-atomistic model, where 4 carbon atoms are represented by 1 CG bead. The CG force-field was developed based on a strain energy conservation approach,34, 55 and it includes bonded contributions from bonds 𝑉𝑏 , angles 𝑉𝑎 , and dihedrals 𝑉𝑑 , and nonbonded interactions 𝑉𝑛𝑏 . The CG force-field parameters are calibrated using mechanical properties obtained from density functional theory and experiments, such as elastic tensile and shear modulus, and failure properties. The hexagonal symmetry of the atomic lattice is thus conserved, which is important to capture the shear response and interlayer adhesion energy, including anisotropic shear stiffness in the zigzag vs. armchair directions and the superlubricity effect. The detailed description of the CG model can be found in the previous work.34 All CG-MD simulations in our study are carried out using the LAMMPS software package.56 To simulate a graphene melt, we first generate a single ribbon-like graphene sheet having a dimension of about 8 nm in width and 48 nm in length, which is sufficiently large comparing to the thickness ( ≈ 0.335 nm) for representing a sheet-like structure. The selection of the ribbon-like sheet geometry and size in this study allows for computational expediency and serves as a representative one. The influence of the sheet size and geometry on the structural and dynamic properties deserves to be explored in future work. Each individual sheet consists of 3 940 CG beads. Then, a total number of 40 sheets are packed into the simulation box with random orientations. Periodic boundary conditions are applied in all the dimensions. Consistent with our prior work, a timestep of ∆𝑡 =⁡ 4 fs is applied. An energy minimization is performed using the conjugate gradient algorithm, followed by two annealing cycles using the NPT ensemble with a constant pressure 1


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atm applied by cycling the temperature from 300 K to 5 000 K over a period of 4 ns until the volume and energy of the systems become independent of time. The system is further relaxed at each temperature for additional 0.8 ns before the production run of 2 ns to collect the data. Four sets of simulations with random initial configurations are performed to improve the sampling statistics and quantify the errors of 𝑇𝑔 and shear modulus calculations.

Calculation of the Structural Properties of Graphene Sheets. From the simulation trajectories at an elevated temperature (𝑇 = 3 000 K), we evaluate various structural properties of graphene sheets at different states (i.e., in solution and melt) using the ZENO software,57 which is based on a path-integration algorithm. In brief, ZENO exploits the relationship between the Laplacian operator and random walks, which can be solved using a probabilistic approach. More detailed information about this algorithm can be found in prior study.57 For an isolated sheet, we compute the hydrodynamic radius 𝑅ℎ , the radius of gyration tensor 𝑹𝟐𝒈 , and the intrinsic viscosity [𝜂 ] from 1 600 different configurations of the sheet. Similarly, we also calculate 𝑹𝟐𝒈 for the graphene sheets in the melt. These properties and their fluctuations provide important shape descriptions that can be related to measurements. The hydrodynamic radius 𝑅ℎ for isolated graphene sheets or particles in solution having viscosity 𝜂0 is related to its diffusion coefficient 𝐷𝑝 of the particles in solution by the Stokes-Einstein relationship,

Dp =

k BT 6πη0 Rh

(4)

where 𝑘𝐵 is the Boltzmann constant. Experimentally, 𝑅ℎ is commonly obtained by dynamic light scattering (DLS).58 For a spherical particle having a radius R, the hydrodynamic radius equals its radius, 𝑅ℎ = 𝑅. Continuum hydrodynamics indicates that if particles are added to a fluid that has


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a viscosity 𝜂0 , the resulting fluid will have a higher viscosity 𝜂, which depends on the volume fraction 𝜙 of the added particles. In such system, the intrinsic viscosity [𝜂] is defined by [𝜂] = lim

𝜂−𝜂0

𝜙→0 𝜂0 𝜙

, which depends only on the geometry of the particles at infinite dilution, and therefore,

[𝜂] constitutes a shape descriptor for an isolated graphene sheet. For the particular case of a rigid spherical particle, [𝜂] = 5/2, and for an asymmetric particle, [𝜂] is greater than 5/2.59 The radius of gyration tensor 𝑹𝟐𝒈 is another important measure of particle size and shape and it is defined by,

 Rxx2  Rg2 =  Ryx2  Rzy2 

Rxy2 Ryy2 2 zy

R

Rxz2   Ryz2  Rzz2 

(5)

2 Specifically, the x-y component 𝑅𝑥𝑦 ⁡of 𝑹𝟐𝒈 equals,

Rxy2 =

1 N N  ( xi - x j )( yi - y j ) 2 N 2 i =1 j =1

(6)

where 𝑁 is the number of particles that form the GS and 𝑥𝑖,𝑗 and 𝑦𝑖,𝑗 are the 𝑥 and 𝑦⁡coordinates of the 𝑖, 𝑗-particle, respectively. The radius of gyration 𝑅𝑔 can be obtained from the trace of the diagonal elements of the tensor 𝑹𝟐𝒈 , 𝑅𝑔 = √Λ1 + Λ 2 + Λ 3, where Λ 𝑖 are the principal eigenvalues of 𝑹𝟐𝒈 and are ranked in an ascend order, Λ1 ≤ Λ 2 ≤ Λ 3 . From the Λ 𝑖 , we can determine a commonly used shape descriptor 𝜅 2 , called the relative shape anisotropy,

 2 = 1− 3

1 2 +  2  3 +  31 (1 +  2 + 3 )2


(7)

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The value of 𝜅 2 is bounded between 0 and 1. In particular, 𝜅 2 = 0 for spherical particles and 𝜅 2 = 1 for rod-like particles.


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ACKNOWLEDGEMENTS The authors acknowledge support by the National Institute of Standards and Technology (NIST) through the Center for Hierarchical Materials Design (CHiMaD). W.X. and S.K. acknowledge support from the Departments of Civil & Environmental Engineering and Mechanical Engineering at Northwestern University. W.X. gratefully acknowledges the support from the NIST-CHiMaD Postdoctoral Fellowship. S.K. acknowledges the support from an ONR Director of Research Early Career Award (PECASE, award #N00014163175). Supercomputing grants from the Raritan HPC System at NIST and the Quest HPC System at Northwestern University are acknowledged.

ASSOCIATED CONTENT Supporting Information. Supporting Information Available: Additional analysis of finite size effects and nanopore size distribution in the graphene melt systems. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION

Corresponding Author * E-mail: [email protected] (W.X.). * E-mail: [email protected] (S.K.). * E-mail: [email protected] (J.F.D.).


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Structure and Dynamics of a Graphene Melt

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