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Molecular Modeling and Mechanics of Acrylic Adhesives on a Graphene Substrate with Roughness

Understanding the mechanics of amorphous polymeric adhesives on a solid substrate at the fundamental scale level is critical for designing and optimizing the mechanics of composite materials. Using molecular dynamics simulations, we investigate the interfacial strength between graphene and polyacrylic and discuss how the surface roughness of graphene affects the interfacial strength in different loading directions. Our results show that a single angstrom increase in graphene roughness can lead to almost eight times higher shear strength, and that such result is insensitive to compression. We have also revealed that the graphene roughness has modest effect on tensile strength of the interface. Our simulations elucidate the molecular mechanism of these different effects in different loading conditions and provide insights for composite designs.


Introduction
The mechanics of amorphous polymeric adhesives on a solid substrate is critical for interfacial interaction for both natural and synthetic composite materials [1][2][3][4][5][6][7][8]. In nature, they are often found in the forms of biological polymers, such as protein, chitin, and glucose, and they play an important role to integrate different materials of contrasting mechanical or chemical properties, allowing these material building blocks to synergistically assemble and form new composite materials with mechanical functions superior to each of their building blocks [3,[6][7][8]. For example, the shell of nacre is mainly composed of the assembly of polygonal mineral tablets with hundreds of nanometers in thickness that are made of calcium carbonate, and they bind with each other on the contact surface by organic polymeric materials [1]. Molecular nonbonded interactions including van der Walls forces, hydrogen bonding, and electrostatic interactions play the dominating role for adhesion, and it is fascinating that these relatively weak interactions can integrate the mineral tablets to achieve both outstanding stiffness and toughness at the same time, which is usually a drawback for many engineering materials [9]. Learning the molecular deformation mechanisms of adhesives at their interface with solid substrates will be beneficial to generate knowledge and insights that could be helpful to design and tailor the mechanical functions of composite materials.
Graphene as one of the two-dimensional materials that were discovered in recent years has several material advantages for making synthetic composites [10,11]. It is the strongest and stiffest material known so far. Moreover, it is of only single-atom thickness, of good thermal and electrical conductivity, as well as optically transparent. Graphene is naturally discovered in the form of graphite, which is known to have a weak interlayer interaction, and one cannot expect to produce a reliable composite by simply stacking them together [12]. Chemical methods including functionalizing the graphene by decorating chemical groups on the graphene surface can enhance the interfacial interaction by forming covalent bonds but will simultaneously change its mechanical and chemical property [13]. Referring to natural systems such as nacre interface and suture of the bone as shown in Fig. 1a, b, respectively, covalent bonds are rarely used but the structure of the interface plays an important role in governing the interfacial mechanics [1,7,14]. Graphene is known to be extremely flexible, and a small mechanical fluctuation can lead to non-zero roughness because of its out-of-plane deformation such as wrinkle and crumple. It is interesting to ask how the adhesion force between polymeric adhesive and graphene will change with its roughness, as schematically shown in Fig. 1c, which may be useful for both adhesive and composite designs.
Molecular dynamics is an effective tool that has proven extremely useful in elucidating molecular mechanics of polymer materials and its adhesion to other inorganic substrates at the extremely small length scales which currently are not accessible with experimental techniques [4]. Such models have successfully been used to investigate a broad range of polymer systems including basic dynamics and structure of amorphous polymers [15,16], adhesion of end-tethered polymers [17,18], and properties of highly crosslinked network polymer adhesives [19][20][21][22]. Here, we use a simple full atomic molecular dynamics model to directly investigate the extent to which the surface roughness of graphene can affect the adhesion force with a polyacrylic (PA) adhesive. We provide quantitative insights into how the mechanics of adhesive and interfacial structure can synergistically contribute to the interfacial adhesion with graphene which can be used to tune and optimize the adhesive performance and mechanics of graphene-based composites.

Atomistic Polyacrylic Model for Simulations
We employ a simple full atomic model of PA based on the assembly of 60 polymer chains with each of them composed of 30 PA acids with a chemical formula as (C 3 H 4 O 2 ) 30 as shown in Fig. 2a. A flat sheet of graphene without roughness is modeled by a planar hexagonal lattice with a lattice constant of 1.41 Å. This PA chains are initially randomly oriented with at least 40 Å away from the top surface of graphene. The interactions among atoms of PA chains and graphene are described by the consistent valence force field (CVFF) [23], which is a generalized valence force field with parameters provided for amino acids, water, and a variety of other functional groups.
Equation (1) generally represents the total energy (E) of the system that is composed of E bond as the sum of interactions via covalent bonds which is composed of deformation energy of bond lengths, bond angles, torsion angles, and out-of-plane interactions, respectively, as well as E nonbond as the sum of A crocodile skull shows an interface between two neighboring pieces of the bone with a clear zig-zag pattern (photos credit to Z.Q.). c. A schematic of a composite material based on graphene and polymer matrix with their interfaces defined by the surface roughness of graphene (R a ). The loading stress on the interface can be separated into shear stress and tensile stress the van der Waals interactions with a Lennard-Jones function and the Coulombic electrostatic interactions [23]. The model is implemented in the LAMMPS simulation package [24]. The simulation time step is 1 fs. Each simulation is performed in the NVT ensemble (constant temperature of 300 K and constant volume of the simulation system with the periodic boundary condition) in the purpose of getting an equilibrated structure with converged potential energy. Periodic boundary conditions are applied to all directions with a system size of 90 × 50 × 170 Ǻ 3 . The nearest distance between two imaging models (30 Ǻ) along the z direction is set to be far beyond the cutoff distance of van der Waals interactions as 10 Ǻ, and the entire system has zero net charge, and thus, there is no interaction cross the periodic boundary in z direction. We use Ewald summation with relative error criterion of 10 −4 to calculate the electronic interaction as an efficient method to accurately include all the long-distance electrostatics interactions. Visualizations of atomic structures are performed with VMD [25].

Assessment and Creation of Surface Roughness of Graphene
Considering that the most widely used one-dimensional roughness parameter is arithmetic average of absolute height values, the geometry of graphene in our system is characterized for its roughness with a definition as where n is the amount of carbon atoms in graphene and z i is the z coordinate of the atom i. To create a graphene substrate with desired roughness, we simply adopt a sinusoidal function as the targeting geometry. Such targeting geometry has its general function as The schematic of the equilibrium process of the simulation system as the polymer chains subjected to gravity force get deposited toward the graphene substrate. c-e. Graphene of different geometry and roughness is used in this study, which largely determines the equilibrated geometry of the polymer material at the interface, as shown by the simulation snapshots, and these models have roughness of 0, 2.3, and 1.5 Å, respectively. f. The distributions of the density (ρ) of PA at the interface as a function of its distance from the graphene surface for different surfaces as shown in panels c, d, and e, respectively. g. The distribution of the density (ρ Hbond ) of hydrogen bonds at the interface as a function of its distance from the graphene surface. This value is normalized by the bulk state density value (ρ Hbond∞ ) as what is measured at the place far from the substrate surface. h. The radial distribution functions (RDF) of the carbon atoms of a layer of polymer chains with 6 Å in thickness at the bottom of the simulation system (blue curve) and that at the top of the simulation system (red curve). The red curve is shifted by 0.1 to get separated from the other curve. Two simulation snapshots of the carbon atoms in these two layers are inserted  For wavy graphene in one direction, we use L y = ∞ and thus cos(2πy i /L y ) = 1 as the example geometry shown in Fig. 2d. For wavy graphene in the two directions, the example geometry is given as shown in Fig. 2e. Using the definition of roughness given in Eq. (2) and by assuming the atomic area is constant for carbon atoms, the roughness of the sinusoidal surface can be 0 z i target dxdy ¼ 4H = π 2 for the 2D case. It is noted that for the same H value, the 2D case has a smaller roughness with a factor of 2/π than that of the 1D case.
A direct coordinate transformation from planar graphene to this geometry will generate a lot of residue stress in graphene, and thus, we execute a Monte Carlo (MC) simulation to get rid of the residue stress in graphene. To achieve that relaxation, we modify the total energy of graphene to be where λ = 100 kcal/mol/Ǻ 2 is a Lagrange multiplier for the constraint energy of the desired geometry. For each MC step, we randomly selected a single carbon atom and make a random movement trial. This trial leads graphene to a new potential energy state E graphene_t from its old state of potential energy E graphene_0 . The Metropolis criterion is used to decide whether or not to keep the movement or instead revert back to the old configuration with the acceptance probability as where B = 0.01 kcal/mol is used here as an empirical parameter to accelerate the convergence of the total energy. We repeat this MC process for 500,000 steps to make sure to find the equilibrated graphene of different desired geometry as shown in Fig. 2c-e. This MC process is run by Matlab with own codes. It is noted that the graphene with a specific geometry is set to be rigid during following MD simulations in order to maintain its geometry through the PA deposition and loading tests. It is noted that several assumptions are made to the current study based on our full atomic model: 1. we do not consider the effects of defects and functional groups in the current study, and these effects have been addressed in the former studies [26][27][28]. Briefly, the randomly distributed vacancy can homogeneously weaken the interfacial interaction by decreasing the effective amount of contact surface area while such effect will not be strong for defects in the form of dislocations such as 5-7-7-5 or grain boundaries as the atomic area will not significantly change for these defect forms even with out-of-plane wrinkles [26]. Functional groups such as carboxyl and hydroxyl groups can homogeneously enhance the interfacial energy [27,28]. 2. We only consider the uniform roughness in the study as the scale we are focusing on is relatively small. For modeling contact surface with a fulllength scale, there will probably be heterogeneous roughness distribution but the problem will be more efficiently solved by mesoscopic models in the future studies. 3. We are considering that the graphene is interacting with a rigid substrate that is much stiffer than polymer on the other side. To simplify the simulation system and only focus on the effect of roughness on adhesion, we model the entire graphene as a rigid surface to keep its roughness a constant value during the simulation.

Gravity Force for Deposition Simulation
We fully equilibrate the systems in MD simulations to allow the initially randomly distributed PA chains to get freely deposited on graphene surface. Gravity force with acceleration of 10 −4 Kcal/Å/g toward the −z direction is subjected to different compression stress (P). Linear fits are made for the case P = 0 (details shown in Fig. 4) and P = 2200 MPa. σ zxc for the graphene with a 2D wavy geometry (as shown in Fig. 2e) is given individually by stars to get distinguished from the 1D cases applied to all the PA atoms in the system in order to accelerate the deposition and assembly of PA chains. After 5 ns of MD simulation, we find the potential energy converges, and all the PA chains get deposited on graphene surface by adapting to its wavy geometry as shown in Fig. 2c-e. After that, we remove the gravity field and fully relax the system before performing loading tests.

Boundary Conditions of Graphene-PA Interface in Loading
For loading MD simulations, we set graphene as a rigid body and fix all the atoms of PA chains that are more than 20 Å from the mean z coordinate of the graphene. The graphene is moved with a constant velocity of 0.0001 Å/fs in either z axis (tensile as schematic in Fig. 3a) or x axis (shear as schematic in Fig. 4a). The total reaction force and displacement of the graphene are recorded during the test for post analysis.

Results and Discussion
We find that the PA deposited on graphene surface has a unique structure that is very different from its bulk state. The equilibrated graphene-PA system, as shown in Fig. 2c-e, shows an empty layer of space, which is caused by the van der Waals radius of graphene, followed by a much denser layer of PA material at its interface with graphene. This observation is quantified by the density distribution of PA materials on the planar graphene as shown in Fig. 2f, which shows that the PA has a much more varying density value within 15 Å from its interface with graphene. For the case with planar substrate, the two peak values at 3.8 and 8.4 Å from the graphene surface correspond to the center of two dense layers with higher mass density as 180 and 120 % of its bulk state. In comparison, the wrinkled geometries affect the density distribution at the interface by causing the layered structure (closest to graphene surface) to be mixed with the near-bulk state (far from graphene surface but still within the gap) at a same height from the bottom of the system. Although the peaks get less obvious for the wrinkled cases, there are still layered structures at the interface as can be seen in Fig. 2d. Moreover, such layered structure of contrasting density also corresponds to larger density of hydrogen bonds (H bonds) (Fig. 2g), as the first density peak corresponds to an H bond density that is more than four times than that of its bulk state. The singularity of mass and H bond distributions agrees with what has been observed for water at the interfaces with other nanomaterials [29][30][31][32]. In previous studies, it has been shown that a graphene substrate can lead to crystallization of polymer at the interface [33,34] and thus yields enhanced mechanics at the interface [35]. Here, we compute the radial distribution function (RDF) of the carbon atoms of the polymer chains and compare its result for the bottom and the top polymers as shown in Fig. 2h. It is noted that the bottom chains show slightly more organized structure than the top chains but such a difference is not very significant as is probably limited by the small size of our simulation model. These evidences point to the fact that the mechanics of graphene-PA interface can be very different from its bulk state and thus is worth looking into by deforming the interface. The results of tensile tests of the graphene-PA interface are summarized in Fig. 3. With the loading condition schematically shown in Fig. 3a, each stress-displacement curve corresponds to the mechanical response of a graphene-PA interface with a specific graphene roughness value. All of the structures initially feature a similar elastic response, followed by a peak stress (σ c ) and then a drop in the stress which corresponds to the formation of defects and rupture in the material, which is shown by the simulation snapshots in Fig. 3b. The long regime of plastic deformation after the stress peak corresponds to the sliding of polymer chains against each other in loading. The peak stress of each stress-displacement curve of specific surface roughness of graphene is summarized in Fig. 3c with all values normalized by the peak stress of the case of a planar graphene σ c;R a ¼ 0 À Á . As expected, the peak stress increases with increasing roughness due to the extra contact surface area between the graphene and the PA. However, the increase is modest-1-Å increment in R a , which corresponds to a ∼20 % more surface area than its projecting area, corresponds to only 4.5 % increase in σ c . This statement is the same for both onedirectional wavy graphene and two-directional wavy graphene. Moreover, such R a insensitivity is similar for the toughness (γ) of the interface as summarized in Fig. 3d, which corresponds to the cumulative area below the stressdisplacement curve, as 1-Å increment in R a only leads to The effects of surface roughness on the strength of graphene-PA interface with different loading methods 6 % increase in γ. For all of the graphene-PA interfaces considered with different graphene roughness, higher R a overall results in an increase in the peak stress and toughness compared to the planar graphene, but again the effect is small. The mechanism can be obtained from the simulation snapshots as shown in Fig. 3b as the failure of the current interface is mainly governed by the separation of the PA material beyond the closest interface with graphene, which will not significantly change with R a of graphene. Further introduction of crosslinks in the PA material can tune the mechanics of PA, which has been shown in our former study [4], and thus may enhance the effect of roughness on tensile strength and modulus.
The results of shear tests of the graphene-PA interface are summarized in Fig. 4. The test method is schematically shown in Fig. 4a, which also contains the simulation snapshots of different slightly deformed interfaces of specific R a values. Figure 4c provides the stress-displacement curve for the mechanical response of each graphene-PA interface with a specific R a value. All of the structures initially feature a similar elastic response, followed by a yield stress (σ zxc ) and then a plateau region for the ideal plastic deformation of the material in shear deformation. Again, it is noted that the trend is not affected by whether the graphene takes one-directional wavy geometry or twodirectional wavy geometry. It is shown that the surface roughness of graphene has a much stronger effect on the shear stress, as a 1-Å increment in R a leads to 770 % increase in σ zxc , as the linear fit shown in Fig. 5b. The mechanism of this strong effect can be understood by examining the deformation profile of the PA at the interface, which is shown in Fig. 4a, b for three different systems for comparison. It is shown that for the planar graphene, the shear deformation leads to a strong slip boundary between the graphene because of the atomic smoothness of the graphene, and thus, the top layer of PA has a much smaller displacement than the graphene, suggesting that the PA provides small resistance force to keep the graphene from moving. The evidence is given by Fig. 4a as the displacement of each PA atom in the shear direction is given by its appearing color. For graphene with a larger R a value, the slip boundary becomes much less severe as the PA trapped between two graphene surface peaks helps to lock the polymer material to move with graphene and also to dissipate the force along the polymer chain, which is supported by Fig. 4b as much more PA materials are involved in deformation (R a > 0) than the planar case (R a > 0). Moreover, for graphene with a larger R a , the smallest distance between the graphene and the fix boundary gets smaller as we keep the PA material amount and projecting area to be constant, which may also contributed to the shear strength. This argument is supported by comparing the shear strength of the polymer on a 1D wavy graphene and that on a 2D wavy graphene as shown in Fig. 5b; as for the 2D case, a higher shearing strength is obtained than that of the 1D case of a same roughness. Considering the fact that the 2D case needs to have a higher amplitude with a factor of π/2 to reach the same roughness as the 1D case (see BAssessment and Creation of Surface Roughness of Graphene^section for detailed analysis), our results show that not only the roughness value but also the amplitude of wavy geometry should be important to determine the adhesion strength.
Finally, we investigate how the compression stress can alter the shear strength of the interface by repeating the shear tests and applying a normal stress on the graphene as shown in Fig. 5a. We find that the normal stress has only modest effect on the shear strength, as even an extremely strong compression of 2200 MPa can only leads to 170 to 240 % increase in σ zxc , as shown in Fig. 5b. This result suggests that for a fully immersed graphene surface in a polymer matrix, increase of pressure may not be sufficient for a stronger interfacial interaction in shear deformation.

Conclusion
We summarize the main results of our simulations in Table 1. It is shown that the surface roughness of graphene has very different effects on the interfacial strength with PA for different loading conditions. We find that the roughness has a strong effect on shear strength, as by keeping the projecting area and PA amount at a constant value, a single angstrom increase in roughness can lead to 7.7 times stronger shear strength. We reveal that such effect is two orders of magnitude larger than its effect on tensile strength, which is limited by the strength of the non-crosslinked PA chains.
It is noted that the roughness in this study is mainly controlled by the amplitude of the sinusoidal function of the surface geometry. Other factors including wavelength and shape may also alter the result, as well as the mechanics of the PA material, which is governed by its crosslinking state. To evaluate these factors, a full atomic PA model may not be sufficient as it limits the system size and time scales for simulations, and thereby, a mesoscopic model will be more efficient to carry out the following up study to investigate the effect of all these factors.
According to what has been revealed in our study, by creating a rough surface of more complex structural features at the nanoscopic scale level, we can expect to achieve a stronger interfacial interaction between graphene and polymer matrix in shear deformation. As what has been shown in nature such as interface in nacre between mineral plate and suture of the bone. For layered composite materials such as the brick-and-mortar designs [3,36], a stronger interface in shearing means larger resistance in deformation, as well as larger energy dissipation during rupture. Considering these effects in composite design will contribute to both strength and fracture toughness of the composite and may eventually lead innovative composites with improved mechanics.