Mechanistic Origin of the Ultrastrong Adhesion between Graphene

Mechanistic Origin of the Ultrastrong Adhesion between Graphene and a‑SiO2: Beyond van der Waals Sandeep Kumar,* David Parks,* and Ken Kamrin* Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: The origin of the ultrastrong adhesion between graphene and a-SiO2 has remained a mystery. This adhesion is believed to be predominantly van der Waals (vdW) in nature. By rigorously analyzing recently reported blistering and nanoindentation experiments, we show that the ultrastrong adhesion between graphene and a-SiO2 cannot be attributed to vdW forces alone. Our analyses show that the fracture toughness of the graphene/a-SiO2 interface, when the interfacial adhesion is modeled with vdW forces alone, is anomalously weak compared to the measured values. The anomaly is related to an ultrasmall fracture process zone (FPZ): owing to the lack of a third dimension in graphene, the FPZ for the graphene/a-SiO2 interface is extremely small, and the combination of predominantly tensile vdW forces, distributed over such a small area, is bound to result in a correspondingly small interfacial fracture toughness. Through multiscale modeling, combining the results of finite element analysis and molecular dynamics simulations, we show that the adhesion between graphene and a-SiO2 involves two different kinds of interactions: one, a weak, long-range interaction arising from vdW adhesion and, second, discrete, shortrange interactions originating from graphene clinging to the undercoordinated Si (Si·) and the nonbridging O (Si− O·) defects on a-SiO2. A strong resistance to relative opening and sliding provided by the latter mechanism is identified as the operative mechanism responsible for the ultrastrong adhesion between graphene and a-SiO2. KEYWORDS: graphene, a-SiO2, ultrastrong adhesion, mechanochemistry, vdW than those found in typical micromechanical systems,12−14 where the interactions are known to be vdW in nature. This suggests that vdW interactions may not be the dominant mechanism of adhesion at the graphene/a-SiO2 interface and that certain other adhesive mechanisms must also be at play between the two surfaces. These adhesive mechanisms and their origin remain a mystery to date, and in the absence of such knowledge, graphene’s ultrastrong adhesion with a-SiO2 remains an anomaly. Mathematically, the interfacial adhesion between two surfaces is described in terms of a traction−separation law (see refs 15−18 and references therein), which is essentially a functional relation between adhesive traction and separation. For example, a vdW-type interaction, which is believed to be the source of adhesion in typical nanoscale contact systems, can be described by means of a continuum traction−separation relation of the form19−21 (see SI):

G

raphene is endowed with remarkable mechanical and electronic properties that make it an ideal material for micro/nano-electromechanical systems.1−5 Graphene, being atomically thin, must be supportedeither partially or completelyon a substrate when used in a device-related application.4,6,7 One of the most widely used substrate materials in graphene-based devices is amorphous silicon dioxide (aSiO2). a-SiO2 is an excellent dielectric with high chemical stability; it is easily grown on Si wafers via thermal oxidation, and most importantly, it adheres very strongly with graphene.8−10 Koenig et al.8 and Boddeti et al.9 recently measured the interfacial adhesion energy of graphene supported on a-SiO2 using nanoscale blistering experiments and reported an ultrahigh adhesion energy, with the values spanning the range 0.12−0.45 J/m2. The adhesion between the two surfaces is in fact so strong that it virtually clamps the graphene sheet onto the substrate surface, as evident in the nanoindentation experiments of Lee et al.11 The interfacial adhesion in nanoscale contact systems is believed to be predominantly van der Waals (vdW) in nature.1,8−10 However, the measured adhesion energies for the graphene/a-SiO2 interface are orders of magnitude larger © 2016 American Chemical Society

Received: January 17, 2016 Accepted: June 27, 2016 Published: June 27, 2016 6552

DOI: 10.1021/acsnano.6b00382 ACS Nano 2016, 10, 6552−6562

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Figure 1. Schematic of the nanoscale blistering test of Koenig et al.8 Conceptually, in order for the membrane periphery to advance by peeling in the simulations, it is necessary that the loading locally generates an opening force that overcomes the cumulative adhesive forces within the fully developed fracture process zone of radial extent Δ.

t = σvdWζ ̂ + τvdWη ̂

so that Γ0 = ∫ ∞ d0 σvdW(ζ) dζ = ΓvdW. Most of the nanoscale experiments characterizing the graphene/a-SiO2 interfacial adhesion have, therefore, been primarily limited to measuring Γ0 only.8−10 As such, Γ0 is a valid measure of interfacial adhesion; however, the value of Γ0 alone cannot provide mechanistic insights into the various interactions that constitute the interfacial adhesion between graphene and a-SiO2. To gain mechanistic insight into the interfacial adhesion between graphene and a-SiO2, in this work we employ a multiscale modeling-based approach, which combines the results of continuum and atomistic simulations, along with analyses of a variety of nanoscale adhesion/contact experiments. Using such a multiscale modeling approach, we aim to address the following issues: (1) are there, in addition to vdW, significant non-vdW interactions between graphene and a-SiO2; and (2) if such non-vdW interactions indeed exist, what is their mechanistic origin?

(1)

where ζ̂ and η̂ are the unit vectors along the normal and a tangent to the interface, respectively. The normal component of the adhesive traction, σvdW, varies with separation ζ as ⎡⎛ d ⎞10 ⎛ d ⎞4 ⎤ ∂ΦvdW (ζ ) σvdW(ζ ) = − = (9ΓvdW /2d0)⎢⎜ 0 ⎟ − ⎜ 0 ⎟ ⎥ ⎝ ζ ⎠ ⎥⎦ ∂ζ ⎢⎣⎝ ζ ⎠ (2)

where ΦvdW(ζ) is the vdW adhesive potential as a function of separation, ζ, and ΓvdW is the work done against vdW adhesion in taking the two surfaces apart from the equilibrium separation d0 to infinity, i.e., ΓvdW = |ΦvdW(d0) − ΦvdW(∞)| (see SI). Jiang et al.22 showed that a sliding motion does not cause any change in vdW adhesion potential between two surfaces, and this is true irrespective of whether the substrate surface is flat or wavy and whether or not the graphene sheet conforms to the wavy structure of the substrate. Consequently, the interfacial shear strength associated with vdW adhesion is negligibly small, i.e., τvdW = −∂ΦvdW/ ∂η ≈ 0. Different kinds of adhesive mechanisms give rise to characteristically different functional relations between adhesive traction and separation. Therefore, if, in addition to vdW adhesion, there are any non-vdW interactions between graphene and a-SiO2, such interactions must be visible in the traction−separation (t−s) law of the interface, provided the t−s law for the interface could be reliably determined. For example, molecular dynamics (MD) simulations of Gao et al. showed that a layer of H2O molecules at the graphene/a-SiO2 interface could give rise to an added long-range interaction between the two surfaces, which manifests itself in the form of a slowly decaying tail in the t−s relation of the interface.23,24 Unfortunately, owing to atomic thinness and extreme flexibility of graphene, the task of reliably measuring the t−s relation for the graphene/a-SiO2 interface and resolving the constituent interactions constituting the adhesion between the two surfaces from such measurements remain extremely difficult. The total adhesion energy Γ0 is relatively straightforward to measure, and the value equals the area under the traction− separation relation. For the moment we restrict our attention to the consequences of assuming that the adhesion at the graphene/a-SiO2 interface is entirely due to vdW interaction

RESULTS AND DISCUSSION Koenig et al.8 measured the interfacial adhesion between graphene and a-SiO2 via a nanoscale blistering experiment, a schematic of which is shown in Figure 1. The test comprises a cylindrical microcavity of radius a = 2.5 μm and depth h = 250 nm, sealed on top by a graphene sheet. Using nitrogen (N2) gas, the sealed cavity is subjected to a rapidly applied initial pressure differential ΔP0 across the graphene sheet. As a result, the sheet bulges out, forming a raised blister of central height δc. Since graphene is impermeable to gases,25 the number of gas molecules in the cavity remains essentially constant over the time scale of the experiment. During blister formation, the total volume of the enclosure (sealed cavity’s + blister’s) increases, and its internal pressure decreases. The pressure difference across the membrane in the bulged configuration is given by ΔP = ΔP0Vcavity/(Vcavity + Vblister), where Vcavity (=πa2h) and Vblister are the volumes of the cavity and the blister, respectively. Koenig et al. noted that for sufficiently small charging pressures the graphene sheet initially remains adhered to the substrate at the cavity periphery, and δc smoothly increases with ΔP. Beyond a critical pressure difference, ΔPCrit = 1.14−1.25 MPa, the membrane peels away from the substrate surface, extending the crack front to radii greater than a. Both the stretching and 6553


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Figure 2. (a) Central deflection (δc) versus charging pressure-difference (ΔP0) curves obtained from FEA simulations for some of the element sizes considered; all simulations employ a common, experimental value of Γ0 = 0.45 J/m2. For comparison, we have also shown the experimental data (shown as black dots).8 The sudden jumps in the plot denote the onset of delamination of graphene from the substrate. (b) Reaction force (Rs) acting on the substrate as a function of charging pressure difference for the three element sizes considered in (a). The bumps in the curve denote the onset of delamination. (c) Trend showing the variation of critical pressure difference with element size. The dashed line is a polynomial fit to the FEA data.

A stretching-dominated delamination, in general, involves a mix of mode-I and mode-II conditions. For this type of delamination, we obtain the size of the fully developed FPZ as (see SI for details and refs 28 and 29)

the bending caused by the applied pressure loading drive the process of delamination. Delamination (also known as interfacial fracturing) of an elastic laminate adhered to a substrate surface is similar to opening of a crack. During growth of such a crack, the energy release resulting from the crack opening is confined within a certain region ahead of the crack tip, called the fracture process zone (FPZ). The FPZ can be seen as a transitory region separating the fully adhered regime from a fully separated regime. For sufficiently thick laminates (i.e., ones whose thickness is comparable to lateral dimensions), delamination does not involve a significant bending, and the size of the fully developed FPZ is solely an interfacial property; that is, it is determined entirely by the stiffness, the strength, and the adhesion energy of the interface.17,18 However, ultrathin laminates (i.e., ones whose thickness is much smaller than its lateral dimensions) constitute a special case. For example, during a bending-dominated delamination, an ultrathin laminate sharply bends away from the substrate, as shown in Figure 1. Such bending localizes the associated strain energy to a tiny region ahead of the crack tip. The size of this region is determined by the local curvature of the sheet (κ) at the crack tip, which, in turn, is determined by the strength of the interfacial adhesion as well as the bending rigidity + of the sheet. Consequently, in ultrathin laminates, the length of the fully developed FPZ in a bending-dominated mode-I delamination, Δb, is both an interfacial and a structural property. Assuming that vdW interaction alone dominates the t-s relation, an estimate of the critical FPZ size under predominant bending is given by (see SI for details; also see Bao and Suo26 and Yang and Cox27) ⎡ 72Γ + ⎤1/4 Δb(vdW) = ⎢ max0 2 ⎥ ⎣ (Σ vdW ) ⎦

Δs(vdW) =

×

2Y Σmax vdW

⎡(1 − cos α)2 + ⎣

2ΓvdW (sin 2 Y 2

1/2 α + cos2 α /Λ vdW )⎤⎦ − (1 − cos α)

(sin α + cos2 α /Λ vdW )

(4)

where α is the peeling angle, Y (≈354 N/m) is the in-plane stretching modulus of graphene, and ΛvdW is the ratio of vdW toughness in mode II (ΓII) to that in mode I (ΓI = ΓvdW). Because of low shear strength of vdW forces, the mode-II fracture toughness ΓII for vdW adhesion is negligible compared to ΓI, and accordingly, we expect that the toughness ratio for vdW adhesion, ΛvdW, should be vanishingly small. In the limit of ΛvdW → 0, the size of the critical FPZ, as given by eq 4, is maximized when α = 90° (see SI). Taking ΓvdW = Γ0, we calculate this maximum value as Δs(vdW) ≈ 4.5 Å ≈ 3lC−C. During a blistering test, both the stretching and the bending caused by the applied pressure loading contribute to the act of delamination, and as we have shown, in each case, the size of the FPZ is atomically small. For the analyses that follow, we take the size of the critical FPZ as ΔvdW = max(Δs(vdW), Δb(vdW)) = Δs(vdW) = 4.5 Å; that is, we assume that the peeling in blistering experiments is stretching dominated and occurs predominantly in mode I. Fracture opening occurs when the applied opening loading overcomes the adhesive forces in the FPZ. For example, referring to Figure 1, a criterion for the interfacial crack opening can be written as πa 2ΔPCrit =

(3)

2π

∫0 ∫0

= 2πa

where Σmax vdW is the strength of the vdW interfacial adhesion. Equation 3 implies that the smaller the bending rigidity of the laminate, the smaller the size of the critical FPZ for the laminate. For graphene, which has an extremely small bending rigidity (+ = 0.225 × 10−20 N-m30,31), we obtain the size of the fully developed FPZ as Δb(vdW) = 3.75 Å ≈ 2lC−C, where lC−C is the interatomic spacing in graphene.

∫0

Δ vdW

Δ vdW

Σ vdW (r′) dr′a dθ

Σ vdW (r′) dr′

(5)

where r′ = r − a and ΣvdW(r′) = σvdW(ζ(r′)), ζ(r′) being the crack-opening profile. The peak of the t−s relation (eq 2) gives the strength of vdW adhesion as Σmax vdW = 1.465ΓvdW/d0. The traction field in the crack-tip region ΣvdW(r′) = σvdW(ζ(r′)) remains an unknown in the absence of the knowledge of the crack profile ζ(r′), and 6554


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∫0

Δ vdW

Σ vdW (r′) dr′ ≈

1 max Σ vdW Δ vdW 2

(6)

Using the result of eq 6 in eq 5, we obtain an expression for the critical pressure difference as ΔPCrit ≈ Σmax vdW

Δ vdW Δ = 1.465ΓvdW vdW a ad0

(7)

Equation 7 implies that the critical pressure difference in a blistering experiment depends not only on the strength of the interfacial adhesion but also on the size of the fully developed FPZ for the graphene/a-SiO2 interface. If the adhesion at the graphene/a-SiO2 interface is entirely due to vdW forces, then we will have ΓvdW = Γ0 = 0.45 J/m2. Taking a = 2.5 μm and the size of the critical FPZ as ΔvdW = 4.5 Å, we obtain ΔPCrit ≈ 0.3 MPa, which is roughly only one-third of the measured ΔPCrit value in Koenig et al.’s experiments. This estimate can be sharpened by employing a nonlinear constitutive model for mechanical deformation of graphene32 and a vdW-based t−s relation for the interfacial adhesion in a finite element analysis (FEA) simulation. We analyze the delamination of a graphene sheet adhered to an a-SiO2 substrate while adopting the same geometric parameters, the kinematic boundary conditions, and the pressure−volume relation as in Koenig et al.’s experiments. Graphene, owing to its extremely small bending rigidity, is modeled by means of membrane elements in the FEA simulations. To achieve an objective (mesh-insensitive) outcome in FEA simulations, the FEA mesh must be able to completely resolve the fully developed FPZ. Because the expected length of the fully developed FPZ for the graphene/a-SiO2 interface measures only three interatomic spacings, completely resolving the fully developed FPZ in an FEA simulation requires an ultrarefine mesh, as is evident in the mesh-refinement studies shown in Figure 2. The ΔPCrit value in the FEA simulations 0 progressively decreases upon a continued mesh refinement, and the objectivity appears to be achieved only in the limit of an ultrarefined mesh, i.e., as le ≪ ΔvdW ≈ O(Å). The objective value of ΔPCrit obtained from the FEA simulations is again 0 anomalously smaller than the measured value of ΔPCrit in 0 Koenig et al.’s experiments. The persistent difference between the FEA analyses and the experimental measurements suggests that the large interfacial toughness of the graphene/a-SiO2 interface noted in the experiments cannot be attributed to vdW-based adhesion alone. Such anomaly is also noticed in continuum analyses of the island blister experiment of Boddeti et al.9 Bodetti et al.’s blister test comprises an annulus-shaped microcavity (a = 1.5 μm, b = 0.35 μm, and h = 112 nm), sealed on top by a graphene sheet, as shown in Figure 3. The microcavity is pressurized by N2 gas to create a rapidly applied initial pressure difference ΔP0 across the membrane, which causes the membrane to bulge out. The bulging causes the pressure difference to change from its initial value to the instantaneous value ΔP(t) = ΔP0Vcavity/(Vcavity + Vblister(t)). For small pressure differences, the graphene sheet remains adhered to the central post as well as to the outer periphery, and the resulting blister is annular in shape (shown by the blue dashed lines in Figure 3). Boddeti et al. reported

Figure 3. Schematic showing island blistering test of Boddeti et al.9 In this case as well, conceptually, we can imagine the onset of delamination happening when the loading overcomes the cumulative adhesive forces in the FPZ.

that as ΔP0 is increased, two instances of delamination are observed: first, an island delamination wherein the graphene sheet lifts off the central post, resulting in a transition of graphene blister shape from annular (blue dashed lines in Figure 3) to spheroidal (red dashed lines). The measured critical pressure difference for this transition is ΔPCrit 0 (island) ≈ 1 MPa.9 Following the delamination from the central post, the membrane delaminates from the outer periphery when ΔP0 reaches a value of ΔPCrit 0 (peripheral) ≈ 2 MPa. On the basis of these transition pressures, Boddeti et al. estimated the adhesion energy for the graphene/a-SiO2 interface as Γ0 = 0.14 J/m2. Using Boddeti et al.’s value of Γ0 = 0.14 J/m2 and the corresponding value of Σmax vdW = 1.465Γ0/d0 in eq 4, we obtain the size of the critical FPZ as ΔvdW ≈ 4.5 Å (same as before). The implication of an ultrasmall FPZ for the onset of delamination in the island blistering tests of Bodetti et al. can be understood by means of a simple analysis. Balance of forces in the vertical direction, referring to Figure 3, gives π (a 2 − b2)ΔP

∫

Δo

∫

Δi

= 2πa Σ vdW (r′) dr′ + 2πb Σ vdW (r″) dr″ 0  0    Fo

Fi

(8)

where r′ = r − a and r″ = r − b; Δi and Δo are the radial extents of the instantaneous FPZs at the inner and the outer periphery, respectively; and Fi and Fo are the reaction forces supported by the FPZs at the inner and outer peripheries, respectively. Employing the analytical solution of Saif et al.,33 the following relation between Fi and Fo is obtained (see SI for details): Fi /Fo =

ln(a 2 /b2)b2 − (a 2 − b2) ≡χ ln(a 2 /b2)a 2 − (a 2 − b2)

(9)

When a > b, the ratio χ between the two reaction forces is Fi. However, on a unit length basis, the reaction force on the inner periphery, i.e., Fi/2πb, is larger than that on the outer periphery, i.e., Fo/2πa. Bodetti et al.9 argued that simultaneous delamination from the island and the outer 6555


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Figure 4. (a) Bulge height (δa) as a function of charging pressure difference (ΔP0) for some of the element sizes considered, all with the common, experimental value of Γ0 = 0.14 J/m2.9 FEA simulations were performed with a systematic mesh refinement in the peripheral region while keeping the mesh size fixed in the island region. (b) Substrate reaction (Rs) as a function of charging pressure difference for some of the element sizes considered. The bumps show the onset of delamination. (c) Trend showing variation of critical pressure difference required for peripheral delamination with element size. The dashed line is a polynomial fit to the FEA data.

Figure 5. (a) Schematic showing the nanoindentation setup of Lee et al.11 During the nanoindentation, the graphene sheet remains clamped to the substrate due to interfacial adhesion, in the absence of which the sheet would collapse radially inward. (b) Load (F) versus depth (δc) response curves obtained from FEA simulation of nanoindentation by adopting different values of interfacial shear strength τ0; also shown is the F versus δc for the case in which the graphene sheet is clamped at the hole periphery, i.e, ur|r=a = 0. The interfacial shear stiffness in all the simulations is G = 100 MPa. For comparison, we also show the F versus δc curve obtained from Lee et al.’s measurements.

respective critical pressure-difference values measured in the experiments of Boddeti et al. The results obtained from FEA simulations of the island blistering experiments are shown in Figure 4. Once again, because the fully developed FPZ for the graphene/a-SiO2 interface is atomically small, the FEA simulations require an ultrarefined mesh to achieve objectivity. The mesh-refinement studies, as illustrated in Figure 4c, show that the critical pressure difference required for the onset of peripheral delamination in FEA simulations progressively decreases as the mesh in the peripheral region is continuously refined. The trend obtained from these calculations suggests that an objective value of ΔPCrit 0 (peripheral) is achieved only in the limit of an extremely fine mesh, i.e., le ≪ ΔvdW ≈ O(Å) . Further, the anticipated objective value is much smaller than the corresponding measured value in Boddeti et al.’s experiments. The blistering experiments8,9 predominantly probe the tensile (also called mode I) response of the graphene/a-SiO2 interface. To probe the interfacial strength of the graphene/aSiO2 under a shear (also called mode II) loading, we analyze the nanoindentation experiments of Lee et al.11 Their nanoindentation tests involved instrumented indentation of a

periphery is energetically not favorable and that island delamination should occur before peripheral delamination. Following this assumption, and again restricting the t−s relation in the FPZs to vdW-based forms, the condition for the onset of island delamination is π (a 2 − b2)ΔPCrit(island) = 2πb

∫0

Δ vdW

Σ vdW (r″) dr″(1 + χ −1 )

(10)

After island delamination, a similar process suggests that peripheral delamination occurs when πa 2ΔPCrit(peripheral) = 2πa

∫0

Δ vdW

Σ vdW (r′) dr′

(11)

By substituting the result of eq 6 into eqs 10 and 11, we obtain the critical pressure-difference values for the onset of island and peripheral delaminations as ΔPCrit(island) ≈ 0.15 MPa (corresponding to a charging value of ΔP0Crit(island) = ΔP Crit (island)(1 + V blister /V cavity ) = 0.179 MPa) and ΔPCrit(peripheral) ≈ 0.175 MPa (ΔP0Crit (peripheral) = ΔPCrit(peripheral)(1 + Vblister/Vcavity) = 0.21 MPa), respectively. These estimates are factors of 5 or more smaller than the 6556


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Figure 6. Structure of a-SiO2 obtained from annealing of an α-quartz crystal in MD simulations. Si atoms are shown in yellow, while the O atoms are shown in red. (a) Highlighted in blue are some of the nonbridging O (Si−O·) defect sites present on the a-SiO2 surface. (b) Highlighted in green are some of the undercoordinated Si (Si·) defect sites present on the a-SiO2 surface.

suspended graphene sheet of diameter 2a = 1 μm by a nanoscale diamond indenter of spheroconical root with radius ρ = 16.5 nm and semicone angle α = 30°, as schematically shown in Figure 5a. Once the central defection takes place, the indentation load exerts a radially inward force on the graphene sheet, subjecting the flat graphene/a-SiO2 interface to a shear stress as well as generating radial tension in the graphene layer. In nanoindentation experiments, the interfacial adhesion between the graphene sheet and the a-SiO2 substrate keeps the sheet from displacing radially inward.11 In fact, the indentation load (F) versus indentation depth (δc) curve obtained from an FEA simulation in which the radial motion of the graphene sheet is completely constrained at the periphery, i.e., ur|r=a = 0, agrees very well with the measured response (see Figure 5b). This confirms that the graphene sheet in the experiments remains virtually clamped at the periphery due to the interfacial forces. Suppression of radial sliding can happen only when the interfacial adhesion has a sufficiently large shear strength. Here again, the assumption that the primary interfacial interaction is vdW-based is problematic because the adhesion arising from vdW forces alone typically has a very low interfacial shear strength. Even if the substrate surface is corrugated, a graphene sheet, owing to its atomic thinness and extreme flexibility, can readily conform to such corrugations.1 Jiang et al.22 showed that the conclusion ∂ΦvdW/∂η = 0 holds true for a graphene sheet supported on a perfectly smooth substrate as well as for a graphene sheet fully conforming to the corrugations of a rough substrate. FEA simulations of the nanoindentation experiment incorporating a nonzero shear strength (τ0) in the adhesion model are also carried out to assess the level of shear resistance required to prevent the inward sliding of the graphene sheet caused by the indentation load. With relatively high (for vdW interaction) τ0 values of 10 MPa and even 50 MPa, inward radial movement of the graphene sheet cannot be prevented in FEA simulations (see SI), and the indention load versus indentation depth curves obtained from such simulations (see Figure 5b) lie substantially below the experimentally measured curve. The shearing resistance associated with vdW forces alone is insufficient to limit the inward sliding of the graphene sheet.

The FEA analyses of blistering and nanoindentation experiments demonstrate that the large tensile toughness and shear toughness of the graphene/a-SiO2 interface measured in and inferred from the experiments simply cannot be explained by a vdW-type adhesion alone. This implies that the adhesion at the graphene/a-SiO2 interface must comprise, in addition to a vdW-type adhesion, certain other adhesive mechanisms as well. In the following section, we suggest alternative adhesive mechanisms that can add to the adhesive tensile and, especially, shearing resistance, compared to a vdW-only interaction. The surface of an a-SiO2 substrate inherently contains defect sites comprising the undercoordinated Si (Lewis structure  Si·) and the nonbridging O (Lewis structure Si−O·) atoms.34,35,37,38 To confirm the presence of such defect species on the surface of an a-SiO2 substrate, we generate an a-SiO2 structure via annealing in MD simulations. The amorphous structure is obtained by initiating a crystalline phase of SiO2 (alpha-quartz) at 6000 K and slowly cooling it to room temperature via a viscous dissipation within an NVE ensemble. The resulting amorphous structure, which is shown in Figure 6, confirms the presence of the undercoordinated Si (highlighted in green) and the nonbridging O (highlighted in blue) defects on the surface. Previous MD simulations37,38 have also shown that concentration of such defects depends upon the thermal history: a rapid cooling results in a higher concentration of surface defects, while a slowly cooled surface contains fewer such defects. Due to high reactivity of the undercoordinated Si and the nonbridging O defects, such defects can strongly influence the adhesion of graphene with a-SiO2. Numerous experimental studies and theoretical calculations have shown that the interaction between graphene and the Si· and Si−O· defect sites on a-SiO2 occurs at the electronic level. For example, measurements have shown that a graphene sheet supported on an a-SiO 2 substrate exhibits a spatial inhomogeneity in electronic density (also known as electron−hole puddles).39 Zhang et al., using a Dirac point mapping technique, examined these electron−hole puddles in a-SiO2-supported graphene and concluded that such fluctuations are caused by graphene’s interaction with certain electrondonating/electron-accepting species on a-SiO2.40 Combining 6557


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Figure 7. (a−c) Snapshots from the MD simulations, showing various stages of peeling of a graphene sheet off an a-SiO2 substrate. Highlighted in the white circles are the bonded interactions between graphene and the undercoordinated Si/the nonbridging O atoms. These defects act as anchoring spots, locally arresting the interfacial crack front. The graphene sheet is temporarily stuck at such anchored spots, causing it to bend sharply (shown in (a)) before it detaches from the spot (shown in (b)). (d) Corresponding time-averaged peeling force ⟨FPeeling⟩ as a function of vertical displacement of the right edge. Periodic ups and downs are the signatures of the “arrested” and the “released” states of the motion of the interfacial crack front. (e) the time-averaged pulling force ⟨FPulling⟩ as a function of lateral displacement, obtained from pull-out simulations. (f) MD-calculated charge distribution (Δq in units of |e|) in the supported graphene sheet, showing a spatial charge inhomogeneity (see refs 39−41).

Figure 8. Anchoring of the crack front at the defect sites results in kinking of the crack front.

that a significant depletion of electronic density takes place from nonbridging O atoms on a-SiO2 due to its interaction with graphene, further corroborating the conclusions of Zhang et al. and Rumero et al. Our proposition is that graphene’s bonding with the Si· and Si−O· defect sites may substantially enhance the interfacial adhesion at the gra-

experimental measurements with theoretical calculations, Rumero et al.41 showed that the nonbridging O defects on aSiO2 indeed act as charge-donating species, and a substantial transfer of electronic charge can take place from such species to a supported graphene sheet. Dispersion-corrected, spinpolarized DFT calculations of Jang et al.42 also confirmed 6558


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Figure 9. Radial distribution functions obtained from MD simulations for (a) the a-SiO2 substrate alone and (b) the graphene/a-SiO2 system.

(RDF) of the graphene/a-SiO2 system. The various peaks in the RDF are related to the various interspecies interactions in the graphene/a-SiO2 system. The RDF of the graphene/a-SiO2 system (shown in Figure 9b) contains a number of distinct additional peaks, which are otherwise absent in the RDF of aSiO2 alone (shown in Figure 9a). Among the additional peaks, the peaks at r = 1 Å and r = 1.46 Å correspond to C−H bonds and C−C bonds, respectively, while the tiny peak at r = 2.4 Å arises due to graphene interacting with the Si· and the Si− O· defect sites on the a-SiO2 surface. Such interactions also result in a spatial inhomogenity of charge in the graphene sheet, as shown in Figure 7f, which is in accordance with the measurements from the Dirac point mapping technique.40 Both the continuum analyses and the MD simulations suggest that the adhesion at the graphene/a-SiO2 interface involves at least two different types of interactions: one, a typical vdW component, and second, a non-vdW component arising from graphene binding with the defect species on the surface of the a-SiO2 substrate. The vdW component of adhesion between graphene and a-SiO2 originates from the electrostatic attraction between the instantaneous dipoles on graphene and the induced dipoles on the a-SiO2 substrate, and vice versa. These pairwise interactions can be summed, and the sum can be subsequently homogenized (assuming the interatomic spacing as the characteristic length scale) to obtain a continuum-mechanical t−s law, such as in eq 2, for the vdW adhesion. However, the non-vdW component of adhesion arising from graphene clinging to the defect species of the aSiO2 substrate cannot be homogenized using the same length scale because the spacing of such defect species on the a-SiO2 substrate is large compared to atomic spacing in either the aSiO2 substrate or graphene. The average distance between reactive surface sites may amount to several multiples of the interatomic spacing. To describe the non-vdW components of the adhesion within a continuum model, we employ a “Jellium-type model” where the non-vdW component of adhesion is treated by means of discrete, spatially localized terms superposed onto the homogenized traction−separation relation for the vdW component of adhesion. Mathematically, the discrete, spatially localized terms in the t−s law are represented by singular Dirac delta functions such that the normal force acting on an area element, extending radially from r0 to r0 + δr and angularly from θ0 to θ0 + δθ, is given as

phene/a-SiO2 interface, leading to the ultrastrong adhesion between the two surfaces as noted in the experiments. To support our proposition, we investigate the interfacial response of a graphene sheet adhered to an a-SiO2 substrate under tensile (mode-I) and shear (mode-II) loadings using MD simulations. The simulations employ an atomically detailed reactive force field (reaxff) (developed by Newsome et al.46) to describe the interactions between various atomic species in the system. This force field, which is based on bond orders and is obtained by parametrization of a large number of highly accurate density functional studies, can capture bond-breakage and bond-formation events during the course of an MD simulation. Owing to an exceedingly small bending rigidity, a graphene sheet subjected to a peeling action bends sharply along the interfacial crack front, forming a region of large curvature (or a crease) localized within an atomically small FPZ (see Figure 1). Experimental and theoretical studies have shown that regions of sufficiently high curvature in graphene, such as a crease, can be chemically highly reactive. The high chemical reactivity is attributed to charge redistribution associated with the locally large curvatures of the mechanical deformation.43−45 It is possible that such high chemical reactivity, localized along the interfacial crack front, may further enhance the bonding between the defect sites on a-SiO2 and graphene, and studies of this are under way. However, the MD simulations of this work were based on potentials that do not specifically address configurations of extreme curvature and thus do not specifically address this issue. MD simulations confirm that along the crack front graphene indeed clings to the discrete defect species (Si· and Si− O·) on the surface of the a-SiO2 substrate, as shown in Figure 7. The clinging causes the graphene sheet to locally anchor with the substrate surface at the location of such defect sites. During delamination, when the propagating crack front encounters an anchored site, it is locally arrested. The arrested state is characterized by a progressively building peeling force, which causes the graphene sheet to sharply bend at the crack front. The local arresting of the crack front at a defect site also results in kinking of the crack front, as seen in Figure 8. As the local adhesive force reaches its maximum value, the pinned crack front location is released, causing a rapid drop in the peeling force and allowing the crack front to resume its progression until it encounters the next anchored sites. Thus, the motion of the tensile crack front is an intermittent motion, comprising the alternate states of “arrested” and “released”. The signatures of graphene clinging to the discrete defect sites on a-SiO2 are also seen in the radial distribution function 6559


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∫r

fn =

0

θ0 + δθ

Γpp I may be much larger, depending upon the areal density of such interactions. Hence, this contribution can result in a large value for the overall adhesion energy Γ0, as noted in experiments. Further, because of the enhancement in the shear strength from these point-to-point interactions, the mode-II fracture toughness of the interface cannot be treated as negligible anymore, and consequently the revised mode-II to mode-I toughness ratio Λ′ = ΓII′/ΓI′ cannot be assumed as vanishingly small. The fact that the graphene/a-SiO2 interfacial adhesion has an added non-vdW component arising due to graphene clinging to the defect sites on the a-SiO2 surface has two important implications. First, the interfacial adhesion between graphene and a-SiO2 should be strongly influenced by the areal density of the defect sites. The surface density of the Si· and the Si− O· defects depends upon the fabrication technique and the processing history of the a-SiO2 surface.34,36 Therefore, depending upon the processing history of the a-SiO2 substrate used, a substantial variation in the measured adhesion energy could be observed across experiments. This is, in fact, in conformation with the large variation in the measured interfacial fracture toughness noted across experiments.8−10 Second, the geometric conformability of graphene with the substrate morphology may have a strong influence on the interfacial adhesion. STM measurements1,48 and theoretical studies49 have shown that a monolayer graphene sheet conforms to a rough substrate surface much more closely than does a multilayer graphene sheet. Thus, a multilayer graphene sheet is likely to come in electronic interaction with much fewer reactive sites compared to a monolayer graphene sheet, suggesting that a multilayer graphene sheet may exhibit a lower adhesion energy, compared to a monolayer graphene sheet, in agreement with the experimental observations.8

Σ vdW (r )r dr dθ

0

∫r

+

∫θ

r0 + δr

0

∫θ

θ0 + δθ

0

∑ f(rm , θm) m

δ1D[r − rm] δ1D[θ − θm] ·ζ r̂ dr dθ r   δ2D[r − rm , θ − θm]

(12)

and the shear force on the area element is given as fs =

∫r

r0 + δr

0

∫θ

θ0 + δθ

0

∑ f(rm , θm) m

δ1D[r − rm] δ1D[θ − θm] ·η r̂ dr dθ r   δ2D[r − rm , θ − θm]

(13)

where the singular term f(rm, θm) δ1D[r − rm]δ1D[θ − θm]/r denotes the point-to-point interaction between an atom in graphene and a reactive site at a location with the coordinates (rm, θm) on the substrate surface, and the sum is over all such point-to-point interactions. δ2D[r−rm, θ−θm] denotes the 2D Dirac delta function such that ∞

∫0 ∫0

2π

δ2D[r − rm , θ − θm]r dr dθ ∞

=

∫0 ∫0

2π

1 δ1D[r − rm]δ1D[θ − θm]r dr dθ r (14)

=1

where δ1D[·] denotes the 1D Dirac delta function. Next, we assume that all point-to-point forces have a common functional dependence on relative separation vector δ. This functional dependence is given by ⎧ f max -(δ)δ ̂ if δ ||̂ ζ ;̂ ⎪ n f(rm , θm) = ⎨ ⎪ f max -(δ)δ ̂ if δ ||̂ η ̂ ⎩ t

(15)

CONCLUSIONS Detailed multiscale simulations combined with analyses of a variety of nanoscale adhesion/contact experiments have been used to identify and quantify the nature of interfacial adhesion between graphene and the a-SiO2 substrate. The continuum FEA simulations, in which the graphene/a-SiO2 interfacial adhesion is described by vdW forces alone, fail to conform to the majority of experimental observations. This implies that the adhesion at the graphene/a-SiO2 interface cannot be attributed to vdW forces alone, in contradiction to the widely believed notion. By combining the results of these continuum FEA simulations with the insights gained from atomistic MD simulations, we shed light on the mechanistic processes that lead to the ultrastrong adhesion between graphene and a-SiO2. We show that the interfacial adhesion between graphene and aSiO2 comprises interactions at multiple length scales: one, a macroscopic component due to vdW forces, and second, microscopic, point-like (spatially localized) interactions due to graphene clinging to discrete undercoordinated Si/nonbridging O sites on the a-SiO2 surface. Subsequent analyses reveal that the strong anchoring effect originating from the microscopic, point-like interactions is responsible for the ultrastrong adhesion between the two surfaces and its high shear strength. The results presented in this work are significant for several reasons. So far it has been believed that the adhesion between graphene and a-SiO2 is predominantly vdW in nature; we have shown that this is not true. Despite the established knowledge about the surface structure of a-SiO2 and the presence of the defect species on its surface, the microscopic, spatially localized

max for all rm ∈ [0, ∞] and for all θm ∈ [0, 2π], where f max n and f t are the strengths of such forces in tension and shear, respectively, and -(δ) = 4e−γ(δ − d0)[1 − e−γ(δ − d0)] (γ > 0) is a Morse function, the magnitude of which, as a function of δ, first increases, reaches the maximum value of unity, and then rapidly decays to zero. The contributions to mode-I and modeII toughness arising from these discrete, point-to-point interactions can be recovered as

ρa

Γ Ipp =

∑ f nmax ∫

∞

ρa

Γ IIpp =

- (δ ) d δ ;

δ = d0

m=1

∑ f tmax ∫ m=1

∞

- (δ ) d δ

δ = d0

(16)

where ρa is the number of the discrete, point-to-point interactions per unit area of the substrate surface. Our proposition is that the vdW adhesion constitutes only a fraction of the overall mode-I fracture toughness. The total mode-I fracture toughness of the graphene/a-SiO2 interface is the sum of ΓppI and ΓvdW, i.e., pp pp Γ′ I = Γ I + ΓvdW = Γ0, and ΓII′ = Γ II

(17)

The adhesion energy associated with vdW-based adhesion, ΓvdW, as measured in typical micromechanical systems12−14 is at least an order of magnitude smaller than the measured adhesion energy in Koenig et al.’s experiments, i.e., ΓvdW ≪ Γ0. However, the contribution from the discrete, point-to-point interactions 6560


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AUTHOR INFORMATION

interactions that result from such ionic sites have remained missing from the description of the graphene/a-SiO2 adhesion. This work incorporates such microscopic interactions in the description to deduce what appears to be an experimentally consistent interfacial adhesion model for graphene supported on a-SiO2. The resulting model not only unravels the mechanistic origin of the unusually large adhesion energies measured in experiments but also offers a physical explanation for the large variation in the measured adhesion energies noted across experiments. Further, the methodologies presented here constitute a multiscale framework for assessing how the microscopic interactions induced by the surface defects on a substrate surface can influence the adhesion between a monolayer and the substrate. Of course, the details of microscopic interfacial interactions may vary depending upon the monolayer and the substrate; however, the framework presented here is general and should be broadly applicable to other monolayer materials, e.g., h-BN, silicene, and 2-D analogues of bulk crystals, such as MoS2, as well.

Corresponding Authors

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

The authors declare no competing financial interest.

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METHODS MD Simulations. The MD simulations were carried out in a LAAMPS atomistic simulator.47 The simulations comprise a graphene sheet, whose edges have been saturated with hydrogen, supported on an a-SiO2 substrate. The system is allowed to equilibrate at 200 K by applying 10 000 MD steps with a Nosè Hover thermostat in an NVT ensemble, and ultimately, the right edge of the graphene sheet is pulled in fixed increments while the substrate is held fixed. Two loading scenarios are considered. In one scenario, termed the “peel-off” test, the edge is pulled upward in increments of 0.615 Å, subjecting the interface to a tensile-type loading. The other scenario, termed a “pullout” test, involves pulling the graphene sheet sideways in increments of 0.1 Å, subjecting the interface to a shear-type loading. Following each loading increment the system is equilibrated via 10 000 isothermal steps and the peel-off/pull-out forces are averaged over these steps. Gao et al.23 recently studied the influence of a thin layer of water molecules at the graphene/a-SiO2 interface on its interfacial adhesion (also see Na et al.24). It was noted that while the presence of a water layer enhances the range of interaction, the adhesion energies remain considerably lower than the measured values. For these reasons, in our MD studies we assumed a dry graphene/a-SiO2 interface without water. FEA Simulations. The FEA simulations were carried out in the commercial finite element software ABAQUS50 using an explicit scheme in a quasi-static sense; that is, the displacements/loads are smoothly ramped up to their maximum value over a long interval of time. The time intervals are chosen large enough so that the kinetic energy always remains within 1% fraction of the total energy. The graphene sheet is modeled as a circular membrane comprising M3D3 and M3D4R elements. The mechanical response of graphene is described by the hyperelastic nonlinear constitutive model of Kumar and Parks,32 implemented in ABAQUS via a vectorized user material (VUMAT) subroutine. This model has been demonstrated to correctly describe the stress−strain response of graphene for the entire range of deformation up to failure. The contact between graphene and substrate/indenter surfaces is treated within a general contact algorithm based on a user-defined interaction function implemented via a VUINTERACTION subroutine.50

ASSOCIATED CONTENT S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.6b00382. Details of the constitutive model, convention for traction−separation relation, derivation of the various analytical expressions, and FEA procedural details (PDF) 6561


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