Superflexibility of graphene oxide

Significance

Bending of a thin plate simultaneously involves contraction and stretching of matter relative to a neutral plane, and tensile rigidity dictates the ability of a thin platelet to be bent. If graphene or graphene oxide (GO) were actually behaving as thin platelets, they would display high bending rigidity. Bending measurements for atomic monolayers remain particularly challenging because of their difficult manipulation. We quantitatively measure the GO bending rigidity by characterizing the flattening of thermal undulations in response to shear forces in solution. The bending modulus is found to be 1 kT, which is about two orders of magnitude lower than the bending rigidity of neat graphene. Amazingly, the high stiffness of GO is associated with an unexpected low bending modulus.

Abstract

Graphene oxide (GO), the main precursor of graphene-based materials made by solution processing, is known to be very stiff. Indeed, it has a Young’s modulus comparable to steel, on the order of 300 GPa. Despite its very high stiffness, we show here that GO is superflexible. We quantitatively measure the GO bending rigidity by characterizing the flattening of thermal undulations in response to shear forces in solution. Characterizations are performed by the combination of synchrotron X-ray diffraction at small angles and in situ rheology (rheo-SAXS) experiments using the high X-ray flux of a synchrotron source. The bending modulus is found to be 1 kT, which is about two orders of magnitude lower than the bending rigidity of neat graphene. This superflexibility compares with the fluidity of self-assembled liquid bilayers. This behavior is discussed by considering the mechanisms at play in bending and stretching deformations of atomic monolayers. The superflexibility of GO is a unique feature to develop bendable electronics after reduction, films, coatings, and fibers. This unique combination of properties of GO allows for flexibility in processing and fabrication coupled with a robustness in the fabricated structure.

Bending of a thin plate simultaneously involves contraction and stretching of matter relative to a neutral plane (1, 2). Because both contraction and stretching are at play, tensile rigidity dictates the ability of a thin platelet to be bent. Actually the resistance against flexion of a thin platelet is quantified by the so-called bending rigidity, which is given in classical continuum mechanics by the following equation: $\kappa =E{h}^3/(12(1-{\nu }^2))$ where E is the Young’s modulus of the material, h its thickness, and v its Poisson ratio (1, 2). At first sight, comparable to thin platelets, graphene and derived monolayers such as graphene oxide (GO), obtained by oxidation of neat graphene (3), are expected to be relatively flexible so that they can be implemented in flexible electronics (prior reduction) and easily deformable coatings, films, and fibers. If graphene (4–7) or GO (7–13) were actually behaving as thin platelets, they would display high bending rigidity. Indeed, measured and calculated values of the Young’s modulus of GO and reduced GO typically range from 200 to 600 GPa, above that of steel, whereas the modulus of neat graphene is about 1,000 GPa (4–13). We note that a thickness has to be defined to deduce the above values of Young’s moduli (14). Even if this definition is not straightforward for atomic monolayers, it is often considered to be about 0.34 nm for neat graphene (4, 6), the interlayer spacing in graphite, and about 0.7–0.8 nm for GO (9, 10, 12, 13). As indicated above, according to thin-plate theory the bending rigidity of atomic monolayers should also depend on their thickness and Young’s modulus. However, atomic monolayers are neither stretched nor compressed when bent. The continuum mechanics picture is not applicable down to an atomic monolayer which can intrinsically be viewed as an individual neutral plane. Nevertheless, the bending rigidity can be determined experimentally or theoretically without necessarily defining a well-given thickness. Indeed, the bending rigidity corresponds to the ratio of the bending moment to the curvature of the platelet, regardless of its thickness (14). It can also be determined in experiments and simulations from analysis of out-of-plane thermal fluctuations (15, 16) without definition of the graphene thickness. The bending rigidity of atomic monolayers involves mechanisms which differ from that involved in tensile deformations or bending of bi- and multilayered systems. In particular, bending of graphene monolayers is dominated by changes of atomic bond and dihedral angles involving multibody interactions beyond two-body interactions of nearest-atom neighbors (14, 17, 18). Considering this unique character, determination of the bending rigidity of atomic monolayers has been the topic of numerous theoretical and computational studies (14, 15, 17–25). It is expected that the bending rigidity of graphene should be on the order of 1–2 eV corresponding to 40–80 kT, where kT is the thermal energy at room temperature. A few experimental measurements of bending rigidity of multilayer systems could be achieved, but measurements for atomic monolayers remain particularly challenging because of their difficult manipulation. The bending rigidity of monolayers has been indirectly deduced from the phonon spectrum of graphite (26). Even if graphene sheets are interacting in graphite, the obtained value, about 1.2 eV, is often considered as an acceptable estimate. Measurements on individual atomic monolayers were achieved by studying buckling instabilities of clamped, and therefore constrained, monolayers (27). The reported value is about 7 eV, corresponding to 280 kT at room temperature. More recent estimates deduced from thermal fluctuations of cantilevers rise up to extremely large values of nearly 10⁵ kT (16). Measurements for GO, the main precursor of graphene-based materials made by solution processing, could not yet be achieved to our knowledge. GO is obtained by oxidation of neat graphene (3) in which the hexagonal lattice of sp2 carbon atoms is partially disrupted. As for graphene, tension and compression of GO involve changes of the length of C–C bonds, explaining thereby the high stiffness of GO, not so different from that of neat graphene. Throughout the present article, “stiffness” means the in-plane tensile/compression modulus. However, GO shows a hexagonal lattice of sp2 carbon atoms (3) that is partially disrupted and that may lead to unexpected mechanical properties regarding bending deformations.

To answer this question, we use an approach that allows the bending rigidity of free and weakly interacting individualized GO monolayers to be measured. This approach consists of analyzing the flattening of thermal undulations of GO sheets in response to minute forces of a shear flow. Considering their atomic thickness, GO monolayers are indeed expected to fluctuate in the absence of any mechanical constraints. This behavior has been reported for highly flexible and liquid systems formed by surfactant or phospholipid membranes (28–30). Because of their liquid nature, surfactant bilayers display an extremely low bending modulus of only a few kT (31, 32). This low value makes their thermal undulations of large amplitude, and sensitive to the stress resulting from a liquid flow. Similar to liquid membranes, GO is known to form lyotropic liquid crystal phases in both aqueous (33–36) and organic solvents (37), even at low concentration due to a very large aspect ratio (38). GO liquid crystals are oblate nematic with long-range orientational order and pronounced spatial correlations along the nematic director, which is why they are often considered as pseudolamellar systems (39). Using the combination of synchrotron X-ray diffraction at small angles and in situ rheology (rheo-SAXS), we unravel several features of the behavior of GO suspensions under flow. Many suspensions of 2D rigid platelets (40–42), clays in particular, exhibit a strong shear-thinning behavior. In rigid systems, this shear-thinning behavior is generally correlated to shear-induced alignment of the particles. Here, we find that in contrast to other known platelet systems (40–42), shear of GO suspensions is associated with an increase of the fraction of aligned particles rather than an enhancement of their orientational order. In addition, coexistence of materials aligned in different directions is observed, and the relative fractions of these differently aligned species vary with the shear rate. More critically, the spacing between GO layers is found to decrease with increasing shear. This behavior confirms that GO layers exhibit thermal undulations of large amplitude at rest or under weak shear. The GO layers are sufficiently large and flexible so that their undulations can be literally ironed out by the shear. This mechanism, which is reminiscent of the suppression of undulations in highly swollen surfactant lamellar phases (28–30), allows us to measure the bending modulus of GO monolayers. Following models first developed for lamellar phases, the bending rigidity is found to be 1 kT. Furthermore, we observe that the present undulations remain sufficiently small so that the layers, even at rest, can be viewed as 2D extended objects rather than crumpled particles (43, 44). The found bending rigidity of GO monolayers compares therefore with the bending rigidity of liquid membranes. Whereas liquid membranes exhibit low in-plane elasticity (45), GO ranks among the stiffest known materials. This unique behavior is discussed by considering the different mechanisms at play in stretched and bent monolayers.

Results and Discussion

Two types of GO materials, homemade GO (HGO) and commercial GO (CGO), have been used in the present study. Details of their preparation are given in Materials and Methods. Complete results for HGO are presented throughout the paper and additional data for CGO are provided in Supporting Information. GO materials are polydisperse in size and shape. HGO sheets display lateral sizes as large as 10 µm, as shown by the scanning electron micrograph in Fig. S1A, with a mean value of 4.3 µm (Fig. S1C). Qualitatively, both HGO and CGO suspensions display a solid-like behavior at rest while they flow above large enough stresses. Such a behavior is quantitatively confirmed by flow curves pictured in Fig. S2A and determined through shear startup experiments at different shear rates. For each value of the applied shear rate, the stress is observed to reach a constant value after a transient period of about 500–1,000 s. The HGO sample behaves as a yield stress fluid, the constitutive equation of which is well described by a Herschel–Bulkley model (46) with a yield stress of about σ_c = 24 Pa. CGO sample similarly behaves as a yield stress fluid with a yield stress σ_c = 72 Pa (Fig. S2B).

Fig. S1.

(A) SEM image of GO flakes (HGO sample) deposited on a silicon wafer. (B) Atomic force microscopy (AFM) image of a GO flake deposited on a silicon wafer. The white line in the image shows the profile along which the thickness of the flake is measured with the AFM. The obtained value is 0.8 nm. (C) The size distribution of the GO flakes (black curve: fit of a log-normal distribution to the data corresponding to the SEM image). The lateral size of the flakes is defined as the diameter of an equal-area circle associated with each flake.

Fig. S2.

(A) The flow curve, shear stress σ vs. the shear rate $\dot{\mathrm{\gamma }}$ of the HGO suspension obtained by independent shear startup experiments. The red line is the best fit of the data by a Herschel–Bulkley model $\sigma ={\sigma }_c+A{\dot{\gamma }}^n$, with σ_c = 24 Pa, A = 31 Pa.sⁿ, and n = 0.3. (B) The flow curve, shear stress σ vs. the shear rate $\dot{\gamma }$ of the CGO suspension. The red line is the best fit of the data by the Herschel–Bulkley model ($\sigma ={\sigma }_c+A{\dot{\gamma }}^n$). The fitting parameters are the following: σ_c = 72 Pa, A = 9 Pa.sⁿ, and n = 0.54.

The high flux of X-ray synchrotron radiation is well suited to reveal the structure of the aforementioned solutions under shear. As depicted in Fig. 1, two measurement configurations are used: the radial one for which the incident beam passes through the center of the Couette cell along the radial direction, perpendicularly to the direction of the flow, and the tangential one for which the incident beam passes tangentially through the center of the cell gap and parallel to the direction of the flow. GO dispersions have been reported to form liquid-crystalline phases with a pseudolamellar organization (34, 39). Similar behavior is observed in some materials made of stiff platelets. Such systems align under shear with the particles oriented parallel to the cell walls, which would correspond to orientation 1 in Fig. 1 [sometimes this orientation is referred to as the “c” orientation (41)]. As seen further, the situation is more intricate for GO solutions. Two-dimensional scattering steady-state patterns of an HGO dispersion flowing under a constant shear stress of 66 Pa are shown in Fig. 2 A and B.

Fig. 1.

Scheme of the Couette cell depicting tangential (T) and radial (R) configurations of irradiation. The X-ray beam is represented by the arrows in both geometries. GO flakes are sketched by flat platelets and the three possible orientations of these flakes are illustrated (labeled 1, 2, and 3). Orientation 1: GO flakes are parallel to the flow direction and to the Couette cell walls. Orientation 2: GO flakes are parallel to the flow direction but perpendicular to the cell walls. Orientation 3: GO flakes are perpendicular to the flow direction and to the cell walls.

Fig. 2.

(A and B) Two-dimensional X-ray patterns of an HGO suspension flowing under a shear stress of 66 Pa. The two patterns are obtained, respectively, in tangential (A) and radial configuration (B). (C–F) X-ray spectra: scattered intensity I multiplied by the square of the wave vector q as a function of q. The intensity I is obtained by integration of a series of 2D patterns recorded at different shear stresses. (C) Horizontal integrations of the tangential pattern (TH). (D) Vertical integrations of the tangential pattern (TV). (E) Horizontal integrations of the radial pattern (RH). (F) Vertical integrations of the radial pattern (RV). Horizontal and vertical integrations are performed within a sector of angular aperture of 45°. Dashed black lines emphasize the peak shifts. “Notches” appearing in the TV or RV curves are artifacts originating from the masked detector areas (green arrays in A and B).

Strongly anisotropic diffraction patterns are obtained in both radial and tangential configurations. The anisotropy of the tangential pattern shows that orientation 1 is largely predominant among the three main possible ones. Indeed, the flakes, even at rest, tend to spontaneously align parallel to the walls of the Couette cell. Such an alignment minimizes the excluded volume of the flakes and can be viewed as an entropic alignment of the particles. It is observed in lyotropic liquid crystals made of disk-like micelles at equilibrium (47, 48).

In radial configuration, orientation 1 is not visible and no anisotropy would be observed if all of the flakes were showing this orientation. Thus, the clear anisotropic signal pattern observed in the radial configuration reveals that a fraction of the flakes is oriented along the configuration labeled “2” in Fig. 1 [sometimes referred to as “a” orientation (41)]. Moreover, the scattered integrated intensity in the radial configuration is 100× smaller than the intensity in tangential geometry. This difference means that orientation 2 is present but not predominant compared with the orientation parallel to the cell walls. A detailed discussion about the degree of orientations of the GO flakes is given in Radial Integrations and Discussion About the GO Flake Orientations and Fig. S3.

Fig. S3.

Radial integrations realized on 2D patterns at different shear stresses in radial (A) and tangential (B) configuration. Raw data are symbolized by open circles and fits with model (77) are represented by the continuous lines. Graphs C and D show the evolution of α and C, the parameters of the fits, with shear stress.

More quantitative analysis is provided in Fig. 2 with the results of azimuthal integrations of the scattered intensity for both radial (R) and tangential (T) configurations for flows under different applied shear stresses. Azimuthal integrations are performed either horizontally (H) or vertically (V) using an aperture angle of 45°. TH, TV, and RV data provide structural information on GO monolayers that are oriented parallel to the flow direction (orientation 1 and 2), whereas the RH integration provides information about any ordering perpendicular to the shear flow (orientation 3). The presence of at least one broad peak in each spectrum is noteworthy. A second-order peak is even observed in the TH integration at a wave vector twice the wave vector of the first-order diffraction peak. The present data are characteristic of a lamellar phase, as previously observed in graphene suspensions at rest (39, 49) and in other inorganic platelet systems (50, 51). This is why the present phase can be considered as a pseudosmectic or pseudolamellar phase in such concentrated regimes. Related more dilute materials could display nematic ordering without strong positional correlations (52). At large wave vectors, Iq² is found to be constant, at least for configurations in which the scattered intensity is large and not potentially erroneously modified by subtraction of the background intensity. This behavior indicates that the flakes can be viewed as 2D extended objects with a fractal dimension of 2 rather than as crumpled particles. This behavior is reminiscent of the behavior of conventional surfactant lamellar (53). The effect of thermal undulations is not seen because of their small characteristic length scales. We note that Iq² is not constant in the RH integration (Fig. 2E). However, the data have to be considered with caution in this configuration because the scattered intensity is very low and likely affected by subtraction of the background intensity. Nevertheless, this subtraction, regardless of the considered configuration, does not affect the existence of peaks resulting from spatial correlations between the particles. Actually such peaks are clearly visible in all configurations. The maxima that Iq² displays in all of the configurations reveal a characteristic average uniform spacing between the aligned flakes. However, in contrast to mineral lamellar phases (40–42), here the wave vector of the peak is sensitive to the shear stress, as schematized by black dotted lines in the different graphs of Fig. 2. The peaks shift toward larger wave vectors with increasing values of the shear stress. By contrast to the TH, TV, and RV configurations, the RH intensity decreases with the shear stress.

This result can be intuitively understood by considering that flakes perpendicular to the flow experience large drag viscous forces and tend to be reoriented along the flow with increasing shear. Other features are less intuitive and arise from distinctive features of GO materials. The average interlayer distance d between the flakes is given by d = 2π/q*, where q* denotes the wave vector at which Iq² is maximum. As a key result, d is found to decrease with increasing shear stress (Fig. 3). Changes of d are quite surprising because the GO concentration is unchanged and no water is expelled from the sample. The phenomenon is observed for both HGO and CGO (Fig. S4). A similar behavior has been observed in highly swollen and flexible surfactant lamellar phases. It originates from a flattening under shear of thermal undulations (28, 54–56). In conventional lamellar phases, long wavelength bending fluctuations induce repulsive interactions that stabilize the membranes. Suppression of undulations is associated with a decrease of the repulsive interactions between the membranes which can lead to instabilities and even to the collapse of the lamellar phase (30). The stability of GO suspensions is essentially due to electrostatic repulsions between the flakes. Therefore, the decrease of their separation cannot be explained by a direct reduction of repulsive interactions. The present flattening reflects in fact an intrinsic elastic response of the flakes that sustains the shear applied to the suspension. This effect is schematized in Fig. 3. As indicated in ref. 30, the volume fraction $\phi$ for flat lamellae is given by $\phi =t/d$, where t is the actual thickness of the membranes and d the layer spacing. When the membranes fluctuate the volume fraction is now given by $\phi =\langle A/d\rangle t/A_p$, where A is the total surface area of the sheets, and A_p is the projected area orthogonal to the mean normal surface. In our experiments, the volume fraction is kept constant and changes of d are only due to changes of A_p. The latter are governed by thermal fluctuations and by their suppression under flow, regardless of the exact origin of the repulsive interactions between the sheets. Following models developed for thermal fluctuations of lamellar phases, it is possible to predict the effect of shear on the flattening of undulations (29, 30) and in particular the evolution of the relative variation of the layer spacing:

$$\frac{\mathit{\Delta }d}{d_0}\mathrm{=}\frac{kT}{24\pi \mathit{\kappa }}\text{ln}\left(1\mathrm{+}\frac{R^6\hspace{0.17em}{\mathit{\eta }}^2{\dot{\gamma }}^2}{24{\pi }^4{\mathit{\kappa }}^2}\right),$$ [1]

where kT is the thermal energy, R the diameter of the flakes, η the viscosity of water, and κ the bending modulus of the flakes. Δd = d₀ − d is the variation of layer spacing with respect to the reference taken at rest d₀. Derivation of Eq. 1 is given in Determination of the Bending Modulus from Shear-Induced Flattening of Undulations and Fig. S5. We note that a related approach has been used in numerical simulations to analyze thermal fluctuations of graphene sheets and deduce thereby its theoretical bending rigidity (15). This analysis is consistent with our experiments except that there is an effective tension arising from shear forces in the present approach. In the absence of tension, the amplitude of simulated graphene fluctuations remains below 1 Å at 330 K (15). Here, at an even lower temperature of 298 K, the spacing between GO flakes varies by almost 15 Å when the material is sheared. Unfortunately, similar simulations for GO have not yet been performed to our knowledge. Those would be particularly interesting for closer comparisons with the present experiments. Undulation fluctuations are here of giant amplitude and hint at a very low value of bending rigidity of GO compared with neat graphene. Measurements at rest could actually not be performed in the Couette cell, which had to be maintained under rotation for averaging purposes of the diffraction intensity. We have taken for the reference a value of d very close to that measured for the smallest stress above the yield stress of the suspensions. For HGO the d measured for a stress of 66 Pa is 17.95 nm and d₀ is taken as 18 nm. For CGO the d measured for a stress of 77 Pa is 17.35 nm and d₀ is taken as 17.4 nm. We shall emphasize that the rest of the discussion is not very sensitive to the choice of d₀. The presently considered values are in good agreement with interlayer spacing distances previously measured on similar systems at rest (39). Variations of d as a function of the shear stress provide therefore an unprecedented opportunity to measure the bending modulus of GO layers. Such a measurement is of course only possible provided that Eq. 1 can fit the experimental data with only two unknown independent parameters, which are the average size of the flakes R and their bending modulus κ. As shown in Fig. 3B, we actually find an excellent agreement between the model and the experimental data (see Fits of Experimental Data with Distinct Values of the Bending Rigidity, Fig. S6, and Table S1). The parameters deduced from the fits are R = (5.5 ± 2.0) μm and κ = (1.0 ± 0.2) kT for HGO, and R = (3.4 ± 0.3) μm and κ = (1.0 ± 0.2) kT for CGO.

Fig. 3.

(A) Sketch of the shear-induced flattening of the GO flakes. (B) Normalized variation of the spacing between GO layers for HGO and CGO materials as a function of shear rate (blue circles experiments for HGO, green squares experiments for CGO). The black lines show fits of the data using Eq. 1. (Inset) Evolution of the smectic distance d with the shear stress σ in the HGO sample (see Fig. S4 for CGO).

Fig. S4.

Two-dimensional X-ray patterns of the CGO suspension flowing under a constant shear stress of 77 Pa. The two patterns are obtained, respectively, in tangential (A) and radial configuration (B). (C–F) X-ray spectra (scattered intensity as a function of wave vector q obtained after integration of a series of 2D patterns recorded at different shear stresses). (C) Horizontal integrations of the tangential pattern (TH). (D) Vertical integrations of the tangential pattern (TV). (E) Horizontal integrations of the radial pattern (RH). (F) Vertical integrations of the radial pattern (RV). Horizontal (respectively, vertical) integrations are done within a horizontal (respectively, vertical) sector of angular aperture of 45°. The notches appearing in the TV or RV curves are artifacts originating from the masked detector areas (green arrays in A and B).

Fig. S5.

Schematic of undulating GO flakes at rest (Left) and under shear (Right). Because of shear the undulations of amplitude h(r) are flattened. The average spacing between the flakes decreases from d₀ to d in response to shear.

Fig. S6.

Relative variation of layer spacing as a function of shear rate. The best fits using Eq. S4 (Eq. 1 in the main text), with κ and R as free parameters, are represented by black lines for HGO and CGO samples. The green and blue dotted lines represent fits using κ = 2 kT for, respectively, CGO and HGO samples. The green and blue dashed lines represent fits using κ = 0.5 kT for, respectively, CGO and HGO samples.

Table S1.

Values of R deduced from fits of experimental data for HGO and CGO samples using fixed values of κ

Values of κ, kT	R for HGO, μm	R for CGO, μm
κ = 0.5 kT	2.3	1.5
κ = 2.0 kT	30	8.5

The corresponding curves of Eq. S4 are shown in Fig. S6.

The fact that reasonable values are found for the average lateral size of the flakes supports the validity of the present approach. Furthermore, the greater value of R found for HGO is also consistent with the origin of the samples. Indeed, particular care was taken for the production of HGO to achieve large flakes. Such care was not taken to our knowledge for the preparation of commercial CGO, which is therefore expected to display a smaller average lateral size.

The same value of κ found within error bars for both HGO and CGO is natural because both materials are chemically similar, and further confirms the validity of the approach. The bending modulus of about 1 kT is almost 2 orders of magnitude lower than the value of neat graphene (14, 15, 17–27). This difference is much greater than the differences of Young’s moduli of GO and neat graphene, which reflects that bending and stretching do not involve the same mechanisms. Stretching is primarily dominated by changes of the length of C–C bonds between neighboring atoms, whereas bending of an atomic monolayer is dominated by changes of bond and dihedral angles (18). Therefore, bending involves multibody interactions beyond nearest-neighbors interactions. The small difference of Young’s moduli between GO flakes and neat graphene suggests that the disruption of the carbon sp2 lattice when graphene is oxidized has a limited effect on mechanical properties as long as interactions between nearest neighbors are considered. However, as reflected by the large difference of bending rigidity, this disruption has much more dramatic consequences when multibody interactions are considered. The present results call for theoretical and computational studies to more deeply understand the above phenomena. Actually, the ultralow bending rigidity of GO compares with the bending rigidity of self-assembled liquid bilayers (31, 32) which involve only physical interactions.

Self-assembled liquid bilayers display an in-plane elastic modulus of 10 MPa (45), whereas GO has a Young’s modulus of several hundred gigapascal. Nevertheless, despite its quasi-liquid state, GO, such as self-assembled bilayers, preserves a nonnegligible rigidity sufficient to form liquid crystalline phases. This picture is further confirmed by the scattering patterns which indicate a fractal dimension of 2, in contrast with previous light-scattering experiments (43) but in agreement with electron micrographs of freeze-fractured materials (44).

Conclusion

Using the high flux of synchrotron radiation, we have shown that the main precursor of monolayer graphene in material processing exhibits unique structural features under flow. GO particles display a pronounced shear-induced alignment, which is not only valuable for the fabrication of flow-aligned structures and devices (57–65) but also for analyzing the giant thermal fluctuations related to the bending rigidity of GO particles. As observed in some self-assembled soft materials (66–68), rheo-SAXS studies reveal that the flow-induced alignment of GO is more subtle than intuitively expected. Depending on the flow velocity, different fractions of materials align along different directions. More importantly, the spacing between the flakes decreases in response to viscous forces. The above phenomena may strongly alter the structure and therefore the resultant properties of materials made by liquid processing. They also allow for measurements of the bending rigidity of GO flakes, which is found to be 1 kT, and comparable with the bending rigidity of liquid self-assembled bilayers. It is known that the aromatic system and changes of dihedral angles play an important role in the bending rigidity of graphenic materials. The superflexibility of GO shows how disruption of this system can dramatically reduce the bending rigidity. Such a superflexibility is expected to be a significant advantage for the development of functional and highly bendable coatings, films, and fibers based on GO flakes (38, 60, 62, 69–75). The unique combination of in-plane stiffness and flexibility of GO allows for versatility in processing and fabrication coupled with a robustness in the fabricated structure.

Materials and Methods

Two different samples of GO solutions are used in this study. The first type of sample, further referred to as HGO, is made from GO produced by the Australian National Fabrication Facility (ANFF) following our previous report (34). Expanded graphite (EG) is first prepared as a precursor of HGO and is mixed with 100 mL sulfuric acid and 5 g potassium permanganate KMnO₄ per gram of EG. The mixture is stirred for 24 h before being cooled down in an ice bath. The addition of 100 mL deionized water and subsequently of 50 mL H₂O₂ at 35 vol % leads to a light-brown solution. The HGO particles are then washed with a solution of hydrochloric acid at 3 vol %, purified by cross-flow dialysis, which fixes the pH in between 4 and 5. Finally, a gentle shaking leads to the exfoliation of the HGO particles in water, with a final HGO concentration of 13 mg/mL The absence of any sonication in the above protocol allows us to produce large GO sheets with lateral sizes as large as 10 µm, as shown by the scanning electron micrograph in Fig. S1A, with a mean value of 4.3 µm (Fig. S1C). At last, the solution is concentrated by two successive centrifugations: a first one is performed under 1,400 × g during 20 min to clean the solution from the remaining aggregates and few layer particles; second, the supernatant is centrifuged again at 50,000 × g during 45 min which produces a suspension at 48 mg/mL, as measured in the dry matter. The second type of sample is a CGO water suspension provided by Graphenea, which contains 4 mg/mL of solid material, among which 95% consists of monolayers. CGO suspensions are further concentrated following the same protocol for HGO materials. This produces a solution concentrated at 54 mg/mL. Assuming a density of 1.8 g/cc for GO materials, it is deduced that HGO and CGO materials have GO volume fractions of 2.7% and 3.0%, respectively. At such large volume fractions, the samples are well above the so-called Onsager isotropic–nematic transition critical volume fraction. This concentration is theoretically given by ${\phi }_{iso}\sim 4\left(t/D\right)$, where t is the particle thickness and D the diameter of the flakes (76). For GO materials with e ∼ 0.7 nm and D ∼ 4 μm, one indeed would expect ${\phi }_{iso}$ to be on the order of only 0.1%.

In this article we have decided to focus on the data obtained with the homemade materials. Data for commercial GO are reported in Supporting Information. Similar behaviors are observed for both types of solutions prepared with different sources of graphene, which confirms the robustness of the present findings.

Rheological Setup.

The experiments are performed at the synchrotron, in a polished polycarbonate Taylor–Couette shear cell (rotating inner cylinder of radius 24 mm, gap e = 1 mm). Rheological data are recorded with a stress-controlled rheometer (Anton Paar Physica MCR 501). Experiments consist of shear startup flows with shear rates ranging between 0.026 and 262 s⁻¹.

SAXS Measurements.

These experiments are performed on the small-angle scattering beamline I22 at Diamond Synchrotron (UK). The incident beam crosses a thin vacuum pipe up to the Couette cell where it goes through the sample. The scattered beam enters another wide vacuum pipe in which is placed the 2D detector (Pilatus P3-2M). The typical size of the beam is 320 (H) × 80 (V) µm. The size of the beam being smaller than the gap of the shear cell, we adjust its position to get an impact in the center of the gap. The wavelength of the incident beam is fixed at 0.999 Å (or 12.4 keV) and the detector is positioned at 9.7 m from the sample so that the accessible range for the wave vector q defined by q = (4π/λ)sinθ, where 2θ denotes the scattering angle, is 0.002 Å⁻¹ <q < 0.15 Å⁻¹. The intensity I scattered by the materials is computed by subtracting the intensity scattered by the sample to that of pure water, using the software suite DAWN provided at the synchrotron. The very high photons incident flux allows the determination of coupled structural and rheological behaviors.

Radial Integrations and Discussion About the GO Flake Orientations

Radial integrations of the 2D patterns provide quantitative information about the degree of orientation of the GO flakes. An example of such integration is given in Fig. S3. Considering that the suspension of GO flakes displays a pseudosmectic order, the intensities are fitted using a model derived from the so-called Onsager orientation distribution (77), namely

$$fsmec\left(\theta \right)=\frac{1}{4\pi }\left(1+C\left[\frac{\alpha \hspace{0.17em}\cosh (\alpha \hspace{0.17em}\cos \hspace{0.17em}\theta )}{\sinh \alpha }-1\right]\right).$$ [S1]

In Eq. S1, α characterizes the width of the angular distribution with respect to the polar axis: It quantifies the degree of orientation of the anisotropic fraction of layers within the system. The higher α, the greater the orientation of the layers is. C (positive number between 0 and 1) is the relative weight of the population of oriented layers, the remaining ones being considered as nonoriented and contributing to an isotropic background. Fig. S3 C and D displays the evolution of α and of C as a function of shear stress in radial and tangential configurations. We first discuss the case of flakes parallel to the flow and to the Couette cell walls (orientation 1 in Fig. 1 of the main text), as this is the predominant material orientation. α is observed to decrease with increasing values of the shear stress (broader peak), which could suggest a counterintuitive loss of ordering under shear. Actually this is not the case, because a large increase of C, corresponding to the fraction of aligned material, is observed at the same time. This behavior can be interpreted as an increase of the amount of aligned flakes and not as a more pronounced alignment of flakes that would already be aligned. This may likely be due to the dispersity in size, shape, and quality of the present particles. The smaller fraction of matter in orientation 2 follows different trends. As for the predominant material aligned in orientation 1, the fraction of aligned material increases with the shear, even if here the fraction of oriented materials is twice smaller than in orientation 1. However, α in this case increases slightly, meaning that the average degree of ordering is weakly improved with shear.

Determination of the Bending Modulus from Shear-Induced Flattening of Undulations

The bending modulus κ is deduced by following models developed for the shear-induced flattening of undulations of surfactant lamellar phases (29, 30).

As sketched in Fig. S5, undulations are suppressed by shear. This induces stretching of the membranes and decrease of their average spacing. The flow actually induces an effective tension of the flakes because of the friction of the solvent on flake undulations of amplitude h(r). The related shear force per unit surface area is given by f_η = q²η $\dot{\gamma }$ h_q², where h_q is the amplitude of an undulation mode of wave vector q in Fourier space. The amplitude of undulations is related to the average layer spacing between the flakes through the following relationship (30):

$$d=\frac{t}{\mathit{\phi }}\left[1+\frac{1}{2}\langle {\left({\nabla }_{\mathrm{\perp }}h\right)}^2\rangle \right],$$ [S2]

where t is the physical thickness of the flakes and ϕ their volume fraction. The mean-square amplitude of gradient of undulations can be derived by considering the free energy of a flexible membrane subject to shear (29, 30):

$$\langle {\left({\nabla }_{\perp }h\right)}^2\rangle =\frac{kT}{{\left(2\pi \right)}^2}{\displaystyle {\int }_{\pi /R}^{\pi /a}\frac{q^2{d}^{2}q}{\mathit{\kappa }{q}^4\mathrm{+}\frac{{\pi }^2{\mathit{\eta }}^2{\dot{\mathit{\gamma }}}^2}{24\mathit{\kappa }{q}^2}}},$$ [S3]

where a is a cutoff atomic length and R the average lateral size of the flakes, with of course a much smaller than R. We note that in the case of lamellar phases, which have indefinite lateral extension, the upper bound is taken as L_p, the typical characteristic distance between collisions of the membranes. By contrast, graphene flakes are individual objects with finite extension of average size R.

Combining Eqs. S2 and S3 allows the separation d between the flakes to be calculated. It is given by

$$\frac{d_0-d}{d_0}=\frac{kT}{24\pi \mathit{\kappa }}\text{ln}\left(1+\frac{R^6{\mathit{\eta }}^2{\dot{\gamma }}^2}{24{\pi }^4{\mathit{\kappa }}^2}\right),$$ [S4]

where d₀ is the spacing at rest or under very weak shear stress. Eq. S4 is used to determine κ and R.

We note that the coupling of undulations and shear forces is expected to increase the macroscopic viscosity of the system as it suppresses thermal undulations. Conversely this effect should contribute to the shear-thinning behavior of GO suspensions at high shear rate.

Fits of Experimental Data with Distinct Values of the Bending Rigidity

The experimental data are fitted to Eq. S4 by fixing the value of κ and by keeping R as a free parameter. This estimation allows a visualization of deviations from the optimal determination. We take κ = 0.5 kT and 2.0 kT, which are values, respectively, smaller and greater than κ = 1.0 kT as determined in the optimal determination. The data have been fitted using the Origin8.0 software using the Levenberg Marquardt iteration algorithm. It is seen in Fig. S6 that deviations from the optimal determination of 1.0 kT lead to inadequate fits. The values of R determined from the fits are given in Table S1.