Structure and Dynamics of a Graphene Melt - ACS Nano (ACS

May 22, 2018 - Materials Science & Engineering Division, National Institute of Standards and Technology , Gaithersburg , Maryland 20899 ... Abstract I...
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Structure and Dynamics of a Graphene Melt Wenjie Xia,*,†,‡,§ Fernando Vargas-Lara,† Sinan Keten,*,§,∥ and Jack F. Douglas*,† †

Materials Science & Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States ‡ Center for Hierarchical Materials Design, §Department of Civil & Environmental Engineering, and ∥Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208-3109, United States S Supporting Information *

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 bulk graphene materials exhibit a fluid-like behavior 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 dramatic slowing down of their dynamics that is likewise similar to ordinary linear polymer glassforming liquids. Bulk graphene materials in their glassy foam state have an exceptionally large free-volume and high thermal stability due to their high Tg (≈ 1600 K) as compared 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 are 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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sized through surface-assisted polymerization of monomers), in which conformations become crumpled and disordered due to 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 the 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 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

ecently, 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 can 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 crystal exfoliation,6 and these have a wide range of properties. 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 (synthe© XXXX American Chemical Society

Received: January 19, 2018 Accepted: May 22, 2018 Published: May 22, 2018 A

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Figure 1. (a) Mapping from the all-atomistic model (gray 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.

Table 1. Functional Forms and Parameters of the Coarse-Grained Graphene Model interaction

functional form

parameters

Vb(d) = D0 {1−exp[a(d − d0)]}2

d0 = 2.8 Å D0 = 196.38 kcal/mol a = 1.55 Å−1 dcut = 3.49 Å θ0 = 120° kθ = 409.4 kcal/mol kφ = 4.15 kcal/mol

bond for d < dcut angle

Va (θ) kθ (θ − θ0)2

dihedral

Vd (φ) = kφ [1−cos(2φ)]

nonbonded

⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ Vnb(r ) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r ⎠ ⎦ ⎣⎝ r ⎠

σ = 3.46Å ε = 0.82 kcal/mol rcut = 12Å

for r < rcut

graphene sheets in solution and in the melt is crucial to develop an extension of structure−property relationships as established for linear polymers. 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 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 1. 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 several hundred-

temperature regime is particularly relevant for energy conversion and for neutron moderator materials for nuclear reactors,29 flammability suppressing additives for polymer materials,30 gas filtration and isotope separation media,31 etc. Since graphene sheets can be considered to be two-dimensional polymers, 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 the 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 liquidlike 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 rubber-like elasticity and rheological response similar to polymers.33 Understanding the structures and dynamics of B

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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 = 3000 K. Panels d, e, and f are the histograms of the normalized number frequency obtained from panels a, b, and c, respectively. The Rh and [η] data exhibit a Gaussian distribution, and the Rg data exhibit a skew normal distribution.

random coils of linear polymer chains. To observe the wrinkling effect in our simulations, we generate 1600 different configurations spanning over a 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 2 panels a−c show the fluctuations of these properties (Rh, Rg, and [η]) as a function of number of configurations. Figure 2 panels d−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 show a skew normal distribution, which can be attributed to the bending rigidity and finite extensibility of the sheet. 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 2),

fold increase in computational speed in comparison to the allatomistic simulations, thus allowing for access to greater spatiotemporal scales. On the basis of 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 a wide temperature range from a high-T regime, where the sheets behave as a fluid, to a low-T regime, where the sheets approach the glassy state. Our simulations show that 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 a characteristic dynamics of a glass-forming liquid over a large T range above the glass-transition temperature (Tg). Below the Tg, the graphene melt transforms 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 2. Summary of Basic Conformational Properties of an Ideal (i.e., Perfectly Flat), Isolated and Collapsed Graphene Sheet, and Sheets in the Graphene Melta

RESULTS AND DISCUSSION Morphology of an Isolated Graphene Sheet versus Melt. In this section, we first explore the morphological characterizations of an isolated graphene sheet in vacuum at elevated temperature (T ≈ 3000 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. At elevated temperature, the thermal fluctuations cause the isolated graphene sheet to wrinkle, an effect similar to the

sheet

⟨Rg⟩ (nm)

⟨κ2⟩

⟨Rh⟩ (nm)

[η]

ideal isolated collapsed melt

14.0 13.7 ± 0.1 2.3 10.7 ± 1.6

0.913 0.894 ± 0.015 0.004 0.582 ± 0.201

8.8 8.7 ± 0.2 3.2

131.2 125.6 ± 17.8 4.7

a

The uncertainty is estimated by the standard deviation of the properties indicated.

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 selfinteractions of the sheet, which leads to the effective onebody 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/RC

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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 sheet. Comparison of the histograms of (e) Rg and (f) relative shape anisotropy κ2 for the graphene sheets in the melt and an isolated graphene sheet, respectively.

sheet (i.e., in a poor solvent), due to a partial screening of excluded volume interactions by the surrounding sheets in the melt. This observation is qualitatively analogous to the solution behavior of linear polymers, in which polymer chain dimensions in the melt are intermediate between those in a good solvent where the chain is swollen and those in a poor solvent where it is collapsed, due to screening of excluded volume interactions by surrounding polymers.40 Similarly, κ2 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 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 2. We also performed additional simulations with much larger systems (i.e., 400 and 2500 sheets with a similar overall density) (see Figure S1 in Supporting Information) to examine for possible finite size effects; Rg and κ2, and the uncertainty determined by the variance of these properties, are not greatly affected by the system size (Rg variance ≲ 1.9 nm and κ2 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-wise nonbonded interactions as a function of temperature. Figure 4 shows that the time-averaged

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. On the basis of 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 2) 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 comprising disoriented graphene sheets, forming the graphene melts. 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) as compared to the isolated graphene sheet discussed above. Figure 3 panels a and b 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 under 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 κ2, called the relative shape anisotropy, by calculating the eigenvalues of R2g. In particular, κ2 = 0 for isotropic spherical particles and κ2 = 1 for linear particles. Figure 3 panels e and f show the frequency distributions of Rg and κ2 for the graphene melt from our simulations, respectively. We then see that the Rg of the 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 D

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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.

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 versus 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 1610 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 conditions for which oxygen is not present to oxidize the graphene. 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 non-Arrhenius 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), Fs(q , t ) =

1 N

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 are 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.

⎛ DT0 ⎞ τα(T ) = τ∞ exp⎜ ⎟ ⎝ T − T0 ⎠

where τ∞, D, and T0 are fitting parameters associated with the 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 that of ordinary linear polymers, T0 is about 1200 K for the simulated graphene melt, which is much higher compared to that of polymers. From the VFT relation, we can estimate Tg by extrapolating the relaxation data to the empirical observation time scale, τα (Tg) ≈ 100 s, where we find Tg to be around 1580 K (with a standard deviation of 88 K) for the graphene melt. This value is much larger than the Tg values of commonly 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 1600 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 5000 K,47 which is much larger than the Tg of the melt. Our result

N

∑ ⟨exp[−iq·(rj(t ) − rj(0))]⟩ j

(2)

(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 2500 K to 6000 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 the graphene melt can be captured by the well-established Vogel−Fulcher−Tammann (VFT) relation:44−46 E

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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

indicates that the bulk graphene materials have great potential for high temperature applications because of their excellent thermal stability and high Tg. 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 the picosecond time scale over which the molecules are caged by their neighbors. Experimentally, ⟨u2⟩ can be measured via a neutron scattering technique. In our simulations, we evaluate the ⟨u2⟩ from the mean-squared displacement ⟨r2(t)⟩ of the CG beads, for which ⟨u2⟩ is defined as the magnitude of ⟨r2(t)⟩ at a time t ≈ 40 ps, corresponding to the localization time scale. 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[u20/⟨u2⟩] with u20 being 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[(u20/⟨u2⟩)α/2], for which 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 ≈ 6000 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, τα(T ) = τr exp[(ur2 /⟨u 2(T )⟩)α /2 − 1]

(3) 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 versus strain at T = 300 K. The fluctuating gray 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 slope). (Inset) G as a 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).

where and τr are the reference values of ⟨u ⟩ and τα at T ≈ 6000 K, respectively. 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 a quantitative relationship between the long-time relaxation dynamics τα and the fast dynamics property ⟨u2⟩ at picosecond time scales. 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 in the LM.49,50 A value of α near 2 is reasonable according to the LM due to the two-dimensional 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 u2r

2

shear simulation at 0 and 0.5 shear strain. We can see that as the shear strain increases, the graphene sheets are more aligned toward the diagonal direction with an angle of 45° relative to the shear direction at 0.5 shear strain. From the shear testing, the shear stress versus strain curve can be obtained as shown in Figure 6b, where the fluctuating gray line shows the original stress output from our simulations and the dark blue 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 F

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ACS Nano 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. 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 1500 K in a nonlinear relationship. This strong temperature dependence of G of the graphene foam is in contrast to the inplane shear properties of an individual graphene sheet, which are found to be weakly dependent on T in the range from 100 K to 2000 K.51 The values of G for the graphene foam is about 3 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, for example, polystyrene and poly(methyl methacrylate) with a shear modulus of around 1000 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 intersheet 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 2000 MPa, depending on the stacking orientations (i.e., commensurate stacking with a higher stiffness and noncommensurate 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 cross-linking the sheets to enhance stress transfer upon loading as reported in recent studies,52,53 which is similar to ordinary 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 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 does not 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 the temperature-dependent local free volume and molecular caging to be somehow different from the glassy polymer and metallic glasses.

melt exhibits the dramatic 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 transforms 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.

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 four 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 versus 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 as compared 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 3940 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 dimensions. Consistent with our prior work, a time step of Δt = 4 fs is applied. An energy minimization is performed using the conjugate gradient algorithm, followed by two annealing cycles from 300 K to 5000 K using the NPT ensemble with a constant pressure of 1 atm applied 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 = 3000 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 a 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 1600 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

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 G

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ACS Nano diffusion coefficient Dp of the particles in solution by the Stokes− Einstein relationship, Dp =

kBT 6πη0R h

ACKNOWLEDGMENTS 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 No. N00014163175). Supercomputing grants from the Raritan HPC System at NIST and the Quest HPC System at Northwestern University are acknowledged.

(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 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 a system, the intrinsic viscosity [η] is defined by η−η [η] = lim η ϕ 0 , which depends only on the geometry of the particles ϕ→0

0

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 59 /2. The radius of gyration tensor R2g is another important measure of particle size and shape and it is defined by ⎛R 2 R 2 R 2 ⎞ ⎜ xx xy xz ⎟ ⎜ 2 2 2 ⎟ 2 R yy R yz R g = ⎜ R yx ⎟ ⎜⎜ 2 2 2⎟ ⎟ ⎝ R zy R zy R zz ⎠ Specifically, the x−y component 2 R xy =

1 2N 2

N

REFERENCES (1) Dean, C. R.; Young, A. F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K. L.; Hone, J. Boron Nitride Substrates for High-Quality Graphene Electronics. Nat. Nanotechnol. 2010, 5, 722−726. (2) Yang, X.; Cheng, C.; Wang, Y.; Qiu, L.; Li, D. Liquid-Mediated Dense Integration of Graphene Materials for Compact Capacitive Energy Storage. Science 2013, 341, 534−537. (3) Lee, J.-H.; Loya, P. E.; Lou, J.; Thomas, E. L. Dynamic Mechanical Behavior of Multilayer Graphene via Supersonic Projectile Penetration. Science 2014, 346, 1092−1096. (4) Stankovich, S.; Dikin, D. A.; Dommett, G. H. B.; Kohlhaas, K. M.; Zimney, E. J.; Stach, E. A.; Piner, R. D.; Nguyen, S. T.; Ruoff, R. S. Graphene-based Composite Materials. Nature 2006, 442, 282−286. (5) Geim, A. K.; Grigorieva, I. V. Van der Waals Heterostructures. Nature 2013, 499, 419−425. (6) Dong, L.; Yang, J.; Chhowalla, M.; Loh, K. P. Synthesis and Reduction of Large Sized Graphene Oxide Sheets. Chem. Soc. Rev. 2017, 46, 7306−7316. (7) Lee, C.; Wei, X.; Kysar, J. W.; Hone, J. Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene. Science 2008, 321, 385−388. (8) Wei, Y. J.; Wang, B. L.; Wu, J. T.; Yang, R. G.; Dunn, M. L. Bending Rigidity and Gaussian Bending Stiffness of Single-Layered Graphene. Nano Lett. 2013, 13, 26−30. (9) Bieri, M.; Nguyen, M. T.; Groning, O.; Cai, J. M.; Treier, M.; AitMansour, K.; Ruffieux, P.; Pignedoli, C. A.; Passerone, D.; Kastler, M.; Mullen, K.; Fasel, R. Two-Dimensional Polymer Formation on Surfaces: Insight into the Roles of Precursor Mobility and Reactivity. J. Am. Chem. Soc. 2010, 132, 16669−16676. (10) Knauert, S. T.; Douglas, J. F.; Starr, F. W. Morphology and Transport Properties of Two-Dimensional Sheet Polymers. Macromolecules 2010, 43, 3438−3445. (11) Bieri, M.; Treier, M.; Cai, J. M.; Ait-Mansour, K.; Ruffieux, P.; Groning, O.; Groning, P.; Kastler, M.; Rieger, R.; Feng, X. L.; Mullen, K.; Fasel, R. Porous Graphenes: Two-Dimensional Polymer Synthesis with Atomic Precision. Chem. Commun. 2009, 6919−6921. (12) Nam, K. T.; Shelby, S. A.; Choi, P. H.; Marciel, A. B.; Chen, R.; Tan, L.; Chu, T. K.; Mesch, R. A.; Lee, B. C.; Connolly, M. D.; Kisielowski, C.; Zuckermann, R. N. Free-Floating Ultrathin TwoDimensional Crystals from Sequence-Specific Peptoid Polymers. Nat. Mater. 2010, 9, 454−460. (13) Abraham, F. F.; Nelson, D. R. Diffraction from Polymerized Membranes. Science 1990, 249, 393−397. (14) Dou, X.; Koltonow, A. R.; He, X. L.; Jang, H. D.; Wang, Q.; Chung, Y. W.; Huang, J. X. Self-Dispersed Crumpled Graphene Balls in Oil for Friction and Wear Reduction. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, 1528−1533. (15) Ma, X.; Zachariah, M. R.; Zangmeister, C. D. Crumpled Nanopaper from Graphene Oxide. Nano Lett. 2012, 12, 486−489. (16) Deng, S. K.; Berry, V. Wrinkled, Rippled and Crumpled Graphene: An Overview of Formation Mechanism, Electronic Properties, and Applications. Mater. Today 2016, 19, 197−212.

(5) R2xy

of

R2g

equals

N

∑ ∑ (xi − xj)(yi − yj ) 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, R g = Λ1 + Λ 2 + Λ3 , where Λi are the principal eigenvalues of R2g and are ranked in ascending order, Λ1 ≤ Λ2 ≤ Λ3. From the Λi, we can determine a commonly used shape descriptor κ2, called the relative shape anisotropy,

κ2 = 1 − 3

Λ1Λ 2 + Λ 2Λ3 + Λ3Λ1 (Λ1 + Λ 2 + Λ3)2

(7)

The value of κ2 is bounded between 0 and 1. In particular, κ2 = 0 for spherical particles and κ2 = 1 for rodlike particles.

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.8b00524. Additional analysis of finite size effects and nanopore size distribution in the graphene melt systems (PDF)

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Wenjie Xia: 0000-0001-7870-0128 Sinan Keten: 0000-0003-2203-1425 Jack F. Douglas: 0000-0001-7290-2300 Notes

The authors declare no competing financial interest. H

DOI: 10.1021/acsnano.8b00524 ACS Nano XXXX, XXX, XXX−XXX

Article

ACS Nano

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DOI: 10.1021/acsnano.8b00524 ACS Nano XXXX, XXX, XXX−XXX