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graphene interfacial viscoelastic dissipation and deformation parameters: A molecular simulation study
Author’s Accepted Manuscript A Quantitative Correlation between Polyethylene/Graphene Interfacial Viscoelastic Dissipation and Deformation Parameters: A Molecular Simulation Study Sousa Javan Nikkhah, Mohammad Reza Moghbeli, Seyed Majid Hashemianzadeh www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(18)30071-X https://doi.org/10.1016/j.ijadhadh.2018.02.032 JAAD2157
To appear in: International Journal of Adhesion and Adhesives Received date: 3 October 2017 Accepted date: 16 February 2018 Cite this article as: Sousa Javan Nikkhah, Mohammad Reza Moghbeli and Seyed Majid Hashemianzadeh, A Quantitative Correlation between Polyethylene/Graphene Interfacial Viscoelastic Dissipation and Deformation Parameters: A Molecular Simulation Study, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2018.02.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Quantitative Correlation between Polyethylene/Graphene Interfacial Viscoelastic Dissipation and Deformation Parameters: A Molecular Simulation Study Sousa Javan Nikkhah1, Mohammad Reza Moghbeli 1*, Seyed Majid Hashemianzadeh2 1
Smart Polymers and Nanocomposites Research Group, School of Chemical Engineering, Iran
University of Science and Technology (IUST), Tehran 16846–13114, Iran 2
Molecular Simulation Research Lab, Department of Chemistry, Iran University of Science and
Technology (IUST), Tehran 16846–13114, Iran
Correspondence to: Tel: +98 21 77240519; Fax:+98 21 77240495.
[email protected] Abstract In this study, the influence of applied tensile stretching rate and temperature on amorphous polyethylene/graphene (PE/G) and polyethylene/amino functionalized graphene (PE/aFG) interfaces were studied via molecular dynamics (MD) simulations. The simulations results indicated that the interfacial adhesion during stretching process was increased with increasing the stretching rate (V) while decreased with increasing the temperature (T). This behavior can be connected to changing the polymer chains friction arisen from inadequate reorientation time, which especially happened at the higher stretching rates and lower temperatures. Furthermore, the amino functionalization of G strengthened both the thermodynamic work of adhesion and interfacial viscoelastic energy dissipation function, (V, T), in the interface deformation process. In fact, relying the chains on aFG surface with stronger interfacial interactions also increased the required energy for the chains to disentangle and reorient on the surface. Moreover, the stretching rate dependency of the value well follows a power law at all the stretching temperatures. The effect of stretching rate and temperature on value was simultaneously
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investigated utilizing the time-temperature superposition principle. Master curves of the simulation data, (aTV), was obtained for PE/G and PE/aFG interfaces on which the horizontal (aT) and vertical shift factor (bT) were calculated using Williams-Landel-Ferry (WLF) equation and Bueche-Rouse theory, respectively. Keywords: Dissipation function, Interfacial fracture, Master curve, Polyethylene, Graphene, MD simulation. 1. Introduction Various carbon nanostructures such as graphene exhibit exceptional mechanical, thermal and physical properties with ability to considerably reinforce polymer matrix in their nanocomposites. The mechanical properties of reinforced polymers with carbon nanoparticles are influenced by interfacial adhesion between nanofillers and polymer matrix due to an effective load transfer at the interface. The structural and dynamic properties of polymer chains at the interface are extremely changed because of the interfacial confinement effects. Polymer chains in nanocomposites have a large interface area with nanoparticles; hence, the interfacial properties has a crucial role on the ultimate mechanical properties of nanocomposites, which arises from the load transferring from the polymer matrix to the nano-filler through the interfacial region in between. Therefore, it is essential to predict the mechanical behavior of interfacial region at different temperatures and stretching rates to tune its performance during deformation [1]. Amico et al. [2] and Huang and Wang [3] showed that the viscoelastic dissipation energy originated from adhesive fracture and friction at polymer/filler (substrate) bond, depends on the thermodynamic work of adhesion (W), applied bond detachment rate and temperature in both the bulk and interface regions.
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The adhesion fracture energy, G, is defined as the energy per unit area required to separate two surfaces from each other. The fracture energy is consisted of the thermodynamic work of adhesion (W) and rheological viscoelastic dissipation function depending on the bond detachment rate (V) and temperature (T) [4, 5]. Maugis and Barquins [5] proposed the G value according to the following equation: ))
(1)
where W is the thermodynamic work of adhesion, φ is the dissipation function, and αT is the William–Landel–Ferry shift factor for time-temperature superposition principle [6]. Gent and Schultz [4] declared that dissipation energy, which is required for fracture of a viscoelastic adhesive bond from an adherent, is remarkably larger than the W. Kadiyala et al. [7] investigated the adhesion fracture energy and failure mechanism of thermoplastic polymer/metal interfaces via lap shear test at high temperatures. The G dependency for different polymer/solid adhesive bonds have been suggested based on experimental results. Maugis et al. [5] performed the peel test of polyurethane layer adhered to a glass substrate. Their results showed that the dissipation energy as a function of crack velocity followed a power-law model with a power-law coefficient of 0.6. Although, the strength of interfacial region can remarkably influence the mechanical properties of nanocomposites and adhesive bonds, it is impossible to characterize and quantitatively study only the polymer/solid interface region using current laboratory equipments. Additionally, different simultaneous molecular fracture mechanisms cannot be easily distinguished from each other during interface deformation. This is where molecular simulation techniques, such as molecular dynamic (MD) simulations, have been considered as valuable tools to visualize a deep insight of molecular orientation and layering at the interface region [8]. Clear dependency of
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simulated interfacial adhesion on the detachment rate and temperature in a molecular scale would explain the interface chain responses to the rheological variables during bond deformation and provide the ability to compare the dissipation energy at the interfacial and bulk regions. MD simulations prepare detailed information about the chemical or physical phenomena occurred at the interface in nanoscale dimensions [1, 8]. The simulations can promote the explanations of experimental data, and open a direction toward tuning the interface strength [8]. In nanocomposites, a strong polymer/nanofiller interfacial area could considerably influence the rheological and mechanical properties, although it is too difficult to determine the role of interface and bulk regions separately. MD simulation would predict the properties of interfacial region separately, which cannot be estimated using experimental tools [8,9]. Some researchers tried to investigate the failure of polymer/substrate interface via MD simulations. However, few works were carried out to investigate the failure behavior of thermoplastic polymer/substrate interfaces [1, 8‒10]. Yuan et al. [1] investigated the interfacial normal strength of three typical polymers, i.e. polyethylene, polyurethane and polystyrene, confined between two graphene single sheets via normal stretching MD simulations. In our previous work, adhesion strength between PE and graphene was calculated using tensile MD simulations [10]. The main goal was to study the effect of polymer chain length, number of polymer chains, and graphene surface functionalization on the fracture behavior and adhesion strength at the interface. In fact, the interface was studied from the molecular ordering, configuration, and energetic point of views during its stretching simulation. For this purpose, the evolution of density profile of the polymer chains at the interface region and the evolution of different potential energy contributions during deformation process were evaluated. Additionally, ordering and configuration were studied via quantifying order parameter and chain disentanglement during the stretching process,
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respectively. However, there are other important parameters such as stretching rate and temperature, which would influence the interfacial adhesion strength due to the viscoelastic behavior of the polymer chains at the interface region. Therefore, it seems a comprehensive study of the interface from the viscoelastic point of view is necessary to deepen insight about adhesion mechanisms at the interface region during its deformation. In addition, the quantitative evaluation of the interface viscoelastic energy dissipation as a function of above-mentioned rheological parameters can determine interface capability to resist against failure or debonding at different deformation conditions. Despite some published simulation works to determine polymer/nanoparticle bond strength [1, 8‒ 10], little attention has been paid to present a quantitative mathematical correlation between the fracture energy and bond detachment variables only in an interfacial region. In the present research work, the interfacial adhesion between polyethylene (PE) and graphene (G) or amiofunctionalized graphene (aFG) has been studied using MD simulation at various stretching rates and temperatures. The goal of this research work is to investigate the interfacial fracture behavior and the dependency of the viscoelastic dissipation energy function (φ) of the interface region to the bond loading parameters, stretching rate and temperature.
2. Simulation details 2.1. Models construction In the present study, PE as a commercial, easily processable and low-cost thermoplastic was selected as a polymer matrix, which its interfacial adhesion on a single sheet graphene was examined. The amorphous cell procedure [8, 10], has been used to produce the atomistic PE model.
energy of PE box was minimized based on the following procedure. The box
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was subjected to an initial energy minimization using Conjugate Gradient algorithm. Thereafter, the polymer was subjected to a NPT (constant particle number, pressure and temperature) simulation at 300 K for 10 ns at the pressure of 0.0001 GPa after a NVT (constant particle number, volume and temperature) equilibration at 300 K for 8 ns. The first NVT simulation was necessary for pre-equilibration and elimination of energy barriers. Afterward, the bulk density of the pure PE and complete equilibration of the simulation box were obtained during the final NPT simulation at atmospheric pressure. The final equilibrated periodic PE box containing 40 PE chains each one with 300 carbon atoms and dimensions of 68.20×68.20×68.20 Å3, exhibits a density value of 0.88 g/cm3, which is in a good agreement with experimental data published pr
[11]. In order to ensure that an
equilibrated periodic model of PE chains had been obtained, glass transition temperature (Tg) of the model in addition to its density was evaluated. The Tg value of a PE model can be calculated according to instant change in the slope of its density-temperature curve. The protocol of Tg calculation was started by the cooling process of the PE model. At the beginning, the polymer model was equilibrated at 600 K. Afterwards, the temperature was decreased from 600 to 50 K during 22 steps. At each step, a 2000 ps NPT simulation was performed for equilibration. In this case, the data sample was a set of data selected from the last 200 ps of the simulation running. The density variation versus temperature of the PE model is illustrated in Figure 1. The approximate Tg value is considered as the temperature once an obvious instant change in the slope of density-temperature curve has been observed. The calculated Tg value of 220 K is in a good agreement to experimental Tg values reported in the range of 190–300 K by some researchers [9]. As expected, the calculated Tg values of PE model should be lower than those of experimental values because of their lower molecular weights in comparison to commercial PE
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samples. Nonetheless, much faster cooling rate in MD simulations in comparison with real applied cooling rates in experiments caused the predicted Tg values be increased [10]. The agreement between the calculated physical properties of PE model, density and Tg values, and experimental results verifies achieving to the equilibrated PE model. A parallel MD code LAMMPS [12] was applied to perform the equilibration and the following simulations. Sun et al. [13–15] for several organic and in-
validated classes. Nose-Hoover thermostat
and Berendsen barostat were used to control the temperature and the pressure, respectively. The Verlet velocity algorithm was applied for integration with 1fs time step. The non-bonded cutoff distance of 12 Å was used. pristine graphene (G) and the amino-functionalized graphene (aFG) with 4.06 NH2 functional groups per nm2 were used as two nano-fillers and adherent substrates. The G model consists of 1824 carbon atoms with the total surface area of 68.66×66.74 Å2. The selection of the aforementioned amino-functional surface density was due to the highest interfacial adhesion obtained with the PE chains, when compared to the other surface densities [8]. In the case of aFG, the amino-functional groups were assumed to randomly attach onto only one surface of the neat graphene by chemical covalent bonding. The functionalization . An energy minimization run of the neat graphene and aFG was performed using a Conjugate Gradient algorithm. To create PE/G and PE/aFG interfaces, G and aFG surface were introduced near the surface of the polymer at a distance less than the cutoff radius, respectively. The simulation interface boxes were further enlarged to 2000 Å in the z direction to assure that the adjoining systems did not
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interact in the z direction, when 3D periodic boundary conditions were applied. First, the joint systems were heated up to 600 K for 1000 ps and then equilibrated at 300K for further 18000 ps using NVT ensemble. The average amounts of results for the last 1000 ps were considered to calculate the energy of the systems. The height of the polymer layer on the G or aFG interface at the equilibrium state was about 68.23 Å. The autocorrelation function of end-to-end unit vector,, was applied to monitor the relaxation processes of the PE chains at the interfaces once these models reached to equilibrium states. u(t) is the end-to-end unit vector for each polymer chain at time t, and u(0) is the corresponding initial value at the beginning. < > denotes an average over all the polymer chains. Figure 2 represents the corresponding versus time (t) curves for the PE chains in both the unfuctionalized and functionalized interfaces. As shown, while the polymer chains progressively lost memories of their initial configurations, decayed from a unity value to zero. As a result, the simulation time was sufficient for equilibration of the PE on the neat G or aFG surface. 2.2. Stretching simulation simulation of the PE/G or PE/aFG uniaxial deformation, the atoms of top region of the polymer layer were kept rigid. This region was about one quarter of the layer height over the graphene surface, which was assumed to be fixed. As shown in Figure 3(a), the top rigid region was moved upward with a prescribed elongational velocity to perform the uniaxial interface deformation.
the top rigid region
along the z-direction [10] The separation 0.5 Å i
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Å step of stretching, the equilibration process was performed for different duration times to achieve different stretching rates. Additionally, the stretching process in each rate was carried out at different temperatures in order to study the effect of both temperature and stretching rate on the fracture energy at the PE/G or PE/aFG interface. Since the applied forcefield in the simulation could not predict the bond breakage, it was assumed that the chains would not break while slip out in a viscous process. The normal stress was calculated applying the Virial theorem [16] all through the only middle moveable region. The stress in each step of deformation was determined by time-average over the latter half of the equilibrium interval and was recorded versus displacement to exhibit the stress-displacement behavior of each interface system.
3. Results and discussions 3.1. Thermodynamic work of adhesion (W) The functionalization of graphene surface was carried out to investigate the effect of different interfacial interactions on the interfacial fracture energy when compared with the neat graphene. The W term, the reversible free energy change for creation of two new equilibrated free surfaces, which quantifies the thermodynamically interfacial adhesion between two surfaces, was evaluated for the PE/G and PE/aFG interfaces according to the following equation [17,18]. ))
(2)
where A and EInt are the contact area and interaction energy between the PE and G or aFG surface, respectively. EG-PE is the total potential energy of the equilibrated PE/G or PE/aFG interface, EG is the potential energy of the G or aFG, and EPE is the potential energy of the neat polymer. According to the simulation results, the calculated W values of the PE/G and PE/aFG interfaces were equal to 175.93±7.23 and 825.55±12.03 mJm-2, respectively. Accordingly, the 9
attachment of amino groups on the G surface seems to considerably improve the dispersive forces and, therefore, interfacial interactions between the PE and aFG surface. Determining the role of W on the fracture energy (G) of the PE/G and PE/aFG interfaces brings about a deep insight about the surface treatment of reinforcement materials. Although strongly depends on the stretching rate and temperature [4, 5], the proper dissipative process can be revealed if the viscoelastic adhesive layer adheres tightly on a substrate with strong interfacial interactions. However, there is a question about the quantitative role of W in the fracture energy at the interfacial region. Furthermore, the stretching rate and temperature dependency of the , only at the interface region, is also another important interest in designing nanocomposite materials. 3.2. Tensile simulation results Figure 3 shows the stretching process of the PE chains in the PE/G and PE/aFG interfaces. In fact, the interface deformation was arisen from the consequent steps of the chain stretching process. The stress-displacement and stress-strain results obtained from tensile simulation of PE/G interface at a given temperature, i.e. 300 K, are indicated in Figure 4. In order to calculate the strain, the displacement was divided by the initial length of the un-rigid part of the polymer layer in each stretching step. As shown, the stress-displacement curve is almost linear at the beginning of the process before yielding is occurred. After the yield point in the post yielding region, the stress starts to decrease because of the PE chains disentanglement. As shown in Figure 4, the yield stress increases with increasing the stretching rate. This increment can be attributed to increasing friction between the polymer chains, which causes more resistance against the deformation. At the lower stretching rates, the role of post-yielding region is prompted because the chains have more time to adopt with each deformation step. In fact, the chains are able to slide, rotate and disentangle to exhibit more viscous behavior. From
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thermodynamic point of view, it can be suggested that for tensile process at high stretching rate, high monomer friction makes constraints at different points throughout the chain. Therefore, lowering the numbers of configurations (Ω), which can be experienced by the chains, reduces the entropy (S) of the system [19]. This reduction originated from system ordering, which makes it difficult for the chains to relax after each stretching step and increases the resistance against chains deformation at high stretching rates. In contrast at low stretching rates, enough time for chains adoption reduces the friction and, therefore, increases Ω. This behavior enhances the chains disordering and increases the S value. Consequently, the chains at each stretching step have opportunity to experience further rotation and slipping in order to relax and reduce their resistance against the applied deformation. Figure 5 shows the snapshots of deformation process for the PE/G interface with different stretching rates just after interface failure. At lower stretching rate, the deformation seems to occur almost throughout the PE layer (Figs. 5a‒ b). On the contrary, the PE chains in high stretching rate separated from the graphene surface. As shown, only some chain segments could deform and the other segments just near the graphene surface remained undeformable (Figs. 5c‒ d). These different behaviors can be related to the above thermodynamic discussion about different chain resistance versus deformation at different stretching rates. The interfacial interactions for the PE/G interface at different stretching rates are similar. However, the applied stress at higher stretching rate could not easily deform the beneath layers of PE due to more friction among the chains and higher resultant resistance against the deformation. Therefore, the stress just transferred throughout PE layer until it reached to PE/G interface contact region. Consequently, an adhesive failure was observed at the interface before the chains were deformed completely. On the other hand, a fairly cohesive failure mode was
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observed at the lower stretching rates because of more accessible time for chains deformation. The applied force at the low stretching rates was able to deform the different layers of PE. Depending on strength of the interface and deformation rate, the failure may take place either at the interface region in contact to graphene surface or in the bulk region. In this case, the PE/G interface experienced different fracture modes based on stretching rate applied in the deformation process. 3.3. Interfacial fracture energy Experimentally, measuring the interfacial fracture energy between polymer and a solid surface especially at micro- and nanoscale is a challenging subject because the polymers are viscoelastic and soft when compared to rigid inorganic solid materials. In the present study, the interface fracture energy was calculated based on the area underneath the stress-displacement curve. The simulation stress-displacement results for the PE/G interface at different stretching rates are shown in Figure 4. The stress-displacement curve at the stretching rate of 0.25 m s-1 exhibits a typical elastic region which is sequentially followed by a yield, disentanglement, and separation region. The elastic modulus, the initial slope of linear part of the curve, and the yield stress increased with increasing the stretching rate. At the higher stretching rate, the disentanglement and separation region are also observed but not dominant as well. One of the primary aims of general line of this research has been to find a mathematic relationship between the viscoelastic dissipation function (φ) at the interface region and debonding variables, i.e. stretching rate and temperature. 3.3.1. Effect of stretching rate The viscoelastic dissipation function can be calculated in terms of G and W according to Eq. 1, that is, φ = (G-W)/W. The G value of the interface at a given temperature and stretching rate
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was calculated from the area underneath of its corresponding stress-displacement curve and W calculated using Eq. 2. Figure 6 indicates the effect of stretching rate on the viscoelastic dissipation function at different temperatures around the glass-transition temperature (Tg) of the PE. As shown, the value increases with increasing the stretching rate in all temperatures. This observation can be attributed to the predominant viscous response at lower stretching rates. At lower stretching rates, the chains have more time to equilibrate and relax after each separation step, and then slip on each other in a liquid-like behavior. This observation can also be described by Deborah's number,
, the ratio of chain stress relaxation time (τr) to observation time
(τp), which characterizes the fluidity of materials. De depends either τr or temperature change of material. For pressure sensitive adhesive De is defined as an equation,
, where V and l
are stretching rate and initial adhesive thickness, respectively. The De value lowers with decreasing V, indicating more viscous behavior of interface over a substrate at the lower stretching rates [20]. However, increasing the stretching rate improved the value due to predominant elastic response which reflected the higher required stress for chain orientation. Increasing the stretching rate increases De, meaning the elastic behavior is the dominant response. In addition, the friction against chains motions and slippage increases because of short available time for chains reorientation (τp). These elastic response and friction at higher stretching rates resulted in higher interfacial fracture energy. Based on the thermodynamic point of view, the free energy of the system in canonical ensemble, i.e. Helmholtz free energy (A), can be defined according to two approaches of state thermodynamic function and partition function (Q) given by the following equations: (3) 13
where U, S and T are internal energy, entropy and absolute temperature, respectively. In addition (4) in which Q can be obtained as follows: ∬
[-
)
]
(5)
where p, q, N and H are momenta, position, number of particles and total energy (sum of kinetic energy K(pN) and potential energy V(rN)), respectively [19]. Since increasing the stretching rate reduces S, the change in A value at constant temperature can be determined by the U value with changing the stretching rate. Increasing the stretching rate at a constant temperature and given same displacement can increase U value due to more system deviation from its equilibrium state. The amount of A increases with increasing the stretching rate. On the other hand, increasing the stretching rate causes the system experiences the states with higher total energies, H (pN, rN), during deformation process. Therefore, the A value increases with increasing the stretching rate because the system undergoes the states with higher total energy throughout the deformation process. The abrupt reduction of value at the very high stretching rates and higher temperatures occurred during the deformation process (Figure 6). It should be mentioned that decreasing at the very high stretching rates above 240 K can be attributed to insufficient time for chain reorientation after each stretching step. Therefore, it would not be possible for the PE chains to adopt well with each new deformation steps. At this condition, the chains would not be able to slide, stretch, rotate and disentangle. In fact, the chains behave more rigidly and cannot endure the applied stress which can decrease the value. Consequently, the simultaneous presence of strong interfacial interactions, chain disentanglement capability and chains friction during deformation of an interface could result in a high value. 14
Figure 6 shows the variation versus V on a log-log scale plot at the temperatures below 320 K. Fitting power-law function to the simulation data at the various temperatures resulted in the coefficient values in the rage of 0.0616‒ 0.2853. The consistency index and power coefficient values of the power-law relations fitted to the simulation results at different temperatures are summarized in Table 1. As shown, the dissipation function has not significantly changed with increasing the stretching rate at the temperatures near the Tg of PE. It is suggested that at these temperatures, the PE still behaves glassy enough without showing significant viscous energy loss during the tensile deformation. The obtained consistency index indicating the elastic behavior (stiffness) of the interface and power-law coefficients are almost similar at these lower temperatures. In contrast, the value considerably increased with increasing the stretching rate at the higher temperatures above 240 K due to the deformation occurring inside the PE chains during the stretching process. The previous research works showed that the viscoelastic dissipation energy of an adhesive bond significantly increased with increasing of stretching rate. The trend of the presented simulation results in a molecular scale is in a good agreement with experimental results published by Gent and Schultz [4] and Kinloch et al. [21]. It should be mentioned that unlike the simulation method, in experimental tests both the polymer bulk layer and interfacial region contribute in the amount of the . In this research work, the MD simulation provides the capability to investigate the deformation behavior and to calculate the only in the PE/G interfacial region. In addition, the type of the considered adhesive in the present work and the experiments are different. Therefore, the difference between the calculated exponent coefficient values of the simulation and those of experimental values is reasonable. Nonetheless, a similar trend in variation of the 15
interfacial fracture energy as a power-law function in both the simulation and experimental research works was observed. 3.3.2. Effect of temperature The debonding temperature influences the chain stretching through varying the thermal fluctuations and internal monomer friction. Figure 6 indicates the effect of temperature on the function of the PE/G interface. As expected, the value has decreased with increasing the temperatures. By increasing the temperature at a given stretching rate, more chains with high thermal energy are accessible. The temperature raising increases Q and lowers the amount of A, which means a lower G value. Figure 7 shows the stress-displacement curves for the PE/G interface stretched at stretching rate of 0.25 m/s and temperatures of 220, 270 and 320 K. The temperature changing exhibited an obvious effect on the stress-displacement behavior of the interface. As expected, the material stiffness reduces with increasing the temperature. The stress-displacement responses at 220, 270 and 320 K show a similar behavior, i.e. an elastic region and a yield point followed by a gradual displacement. However, it is observed that the maximum stress and fracture energy at 320 K is much smaller than those at 220 and 270 K. This implies that the interface strength becomes weaker with temperature increase in the interface. It was assumed that the chains would not break during debonding while could slip out through a viscous process. Additionally, increasing the temperature decreased the consistency index while enhanced the power coefficient. This observation indicates the reduction of interface elastic behavior with more viscous flow by temperature increasing. In addition, De reduces by increasing the temperature due to reducing the τr value. In this state, the polymer interface behaves more viscously.
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3.3.3. Effect of graphene surface functionalization Figure 8 indicates the stretching velocity dependency of for PE/aFG interface at the various temperatures. As shown, an increase in value was observed with functionalizing the graphene surface at the given temperatures. The attachment of amino groups on the graphene surface increased W [8, 10] due to stronger dispersive interactions. Furthermore, the functionalization also influenced the rheological dissipation behavior in the interface (Figure 6 and 8). This behavior can be attributed to more intensive layering effect of the polymer chains on the aFG when compared to the graphene surface. Figure 9 shows the chain reduced density, i.e., the interface chain density to bulk chain density ratio (ρ/ρ0) as a function of the distance from G or aFG) surface. For both the PE/G and PE/aFG interfaces, a sharp density profile peak was observed at the vicinity of the G or aFG surface, which indicated a dense segment layer near the surface and layering behavior of the PE at both the interfaces. The intensities of other peaks were reduced gradually as the distance from the surface increased. As shown, the intensities of the peaks in the PE/aFG interface are higher than those of the PE/G interface, which can be related to the stronger interfacial interaction at the PE/aFG interface. It can be understood that functionalization of the graphene at the interface led to creating more ordered chain layering region in comparison to the neat PE/G interface. Additionally, the functionalization with amino groups increased the required energy to disentangle and deform PE chains in this well ordered region during the elongation process. Furthermore, stronger interfacial interactions cause more constraints for PE chains at the PE/aFG interface. A less numbers of configurations (Ω) and, consequently, a lower S value was experienced by the chains on the aFG surface when compared to graphene surface at the same temperature, stretching rate and displacement. It is more difficult for the chains to relax at the PE/aFG system after each deformation step due to more tight
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binding of the chains to the aFG surface. Therefore, the U and A values of the PE/aFG interface are higher than those of the PE/G one due to more deviation from equilibrium state. The consistency index and power coefficient amount of the fitted power-law functions to the calculated values for PE/aFG interface at each stretching process temperature are summarized in Table 1. As shown, the consistency index values of the functionalized interface is larger than those of the unfunctionalized interface. Stronger interactions between the chains and amino groups on the graphene surface increased the fracture dissipative energy. On the contrary, the power-law coefficient for the PE/aFG interface in a given temperature is smaller than that of the PE/G interface. This means that the change rate of value with the stretching rate at the functionalized interface is slower than that of the un-functionalized interface. In fact, the chemically bonded amino groups on the graphene surface improve the interfacial interactions and adsorb the chains more tightly than the neat graphene surface. Accordingly, it can be suggested that stronger interactions at the PE/aFG interface cause the chains can rely on the aFG surface and then more chains volume can be deformed before interface failure at the higher stretching rates. 3.3.4. Time-temperature superposition Time-temperature superposition is a valuable method to explain the viscoelastic behavior of polymers in a wide range of time and temperature via making a master curve on which the obtained data in different temperatures are shifted on a reference temperature [6]. The effect of stretching rate (time) and temperature on the energy dissipation function was simultaneously investigated utilizing time-temperature superposition principle. The horizontal shift factor values were calculated according to Williams-Landel-Ferry (WLF) equation [6]. )
(6) 18
where αT is the William–Landel–Ferry shift factor and T is the test temperature. The WLF equation is usually used when the Tg is selected as the reference temperature and T is in the range of Tg
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amount of V and decreasing T value. At higher log αT value, more stress is required to disentangle the chains at the interface. 4. Conclusions In the present study, the correlation between the energy dissipation function of the PE/G and PE/afG interfaces and stretching rate and temperature was investigated. Accordingly, at the low temperatures, the interfacial dissipation function did not significantly change with increasing the stretching rate. At the lower temperatures, the PE interface behaved glassier than the high temperatures in the deformation process. On the contrary, at the higher temperatures above the Tg value, a significant change in the dissipation function was observed with increasing the stretching rate due to the higher viscoelastic dissipation associated with the plastic deformation occurred inside the interface. Fitting power law to the simulation data resulted in the power-law coefficient in the range of 0.0616‒0.2853 for the PE/G interface, and in the range of 0.0598‒ 0.1775 for the PE/afG interface. At the given temperature, the power-law coefficient for the PE/aFG interface was smaller than that of the PE/G interface. This means that the variation of the PE/aFG interface was slower than the PE/G interface. The stronger interfacial interactions at the PE/aFG interface provides the possibility for the chains to slip out in a more viscous process, especially at the higher stretching rate. In fact, the higher interfacial fracture energy is achievable when chains exhibit both elastic and viscous behavior during the stretching process. This behavior can be observed for interfaces with stronger interactions, which enable the adhesive chains to disentangle and reorient properly. Therefore, the resultant friction arisen from these chains movements causes the interface to resist further against the deformation. The present research
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work can provide the opportunity to study the interfacial region from the thermodynamic and rheological point of views and design an interface with higher strength for proper load transfer. Although this present research work has been devoted to investigate the interface between an amorphous PE and graphene, it is obvious that the presence of crystalline phase in the polymer structure at the interface can influence on the interfacial adhesion. Therefore, the chains deformation, sliding and rotation may take place with difficulty. This issue and considering semi-crystalline instead of amorphous PE in simulations will be our future research work. Acknowledgment The authors gratefully appreciate the Iranian Nanotechnology Initiative Council and the visepresident for research and technology of Iran University of Science and Technology (IUST) for their partial financial supports.
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References [1] Yuan Z, Lu Z, Yang Z, Sun J, Xie F. A Criterion for the Normal Properties of Graphene/Polymer Interface. Comput Mater Sci 2016; 120:13−20. [2] Amico FD, Carbone G, Foglia MM, Galietti U. Moving Cracks in Viscoelastic Materials: Temperature and Energy-Release-Rate Measurements. Eng Fract Mech 2013; 98:315−325. [3] Huang YH, Wang J. Prediction of Coating Adhesion Loss due to Stretching. Int J Adhes Adhes 2013; 40: 49−55. [4] Gent AN, Schultz J. Effect of Wetting Liquids on the Strength of Adhesion of Viscoelastic Material. J Adhes 1972; 3: 281−294. [5] Maugis D, Barquins M. Fracture Mechanics and the Adherence of Viscoelastic Bodies. J Phys D: Appl Phys 1978; 1:1989−2023. [6] Williams ML, Landel RF, Ferry JD. Mechanical Properties of Substances of High Molecular Weight .19. The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids. J Am Chem Soc 1955; 77: 3701−3707. [7] Kadiyala AK, Bijw J. Investigations on Performance and Failure Mechanisms of High Temperature Thermoplastic Polymers as Adhesives. Int J Adhes Adhes 2016; 70:90−101. [8] Nikkhah SJ, Moghbeli MR, Hashemianzadeh SM. Investigation of the Interface between Polyethylene and Functionalized Graphene: A Computer Simulation Study. Curr Appl Phys 2015; 15:1188−1199. [9] Awasthi AP, Lagoudas DC, Hammerand DC. Modeling of Graphene–Polymer Interfacial Mechanical Behavior Using Molecular Dynamics, Modelling Simul Mater Sci Eng 2009; 17: 015002.
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[10] Nikkhah SJ, Moghbeli MR, Hashemianzadeh SM. Dynamic Study of Deformation and Adhesion of an Amorphous Polyethylene/Graphene Interface: A Simulation Study. Macromol Theory Simul 2016; 25: 533−549. [11] Brandrup J, Immergut EHE, Grulke A. Polymer Handbook. New York: Wiley-Blackwell; 1999. [12] Plimpton SJ. Fast Parallel Algorithms for Short-Range Molecular Dynamics, J Comput Phys 1995; 117: 1−19. [13] Sun H. Force Field for Computation of Conformational Energies, Structures, and Vibrational Frequencies of Aromatic Polyesters, J Comput Chem 1994; 15: 752. [14] Sun H, Mumby SJ, Maple JR, Hagler AT. An ab Initio CFF93 All-Atom Force Field for Polycarbonates, J Am Chem Soc 1994; 116: 2978−2987. [15] Sun H. COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase Applications: Overview with Details on Alkane and Benzene Compounds. J Phys Chem B 1998; 102: 7338− 7364. [16] Tsai DH. The virial Theorem and Stress Calculation in Molecular Dynamics. J Chem Phys 1979; 70:1375. [17] Kisin S, Vukic JB, van der Varst PG, With G, Koning CE. Estimating the Polymer-Metal Work of Adhesion from Molecular Dynamics Simulations. Chem Mater 2007; 19:903‒ 907. [18] Henry DJ, Yiapanis G, Evans E, Yarovsky I. Adhesion between Graphite and Modified Polyester Surfaces: A Theoretical Study. J Phys Chem B 2005; 109: 17224‒17231. [19] Widom B. Statistical Mechanics, A Concise Introduction for Chemists. New York: Cambridge University Press; 2002. [20] Reiner M. The Deborah Number. Physics Today 1964; 17: 62.
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[21] Kinloch AJ, Lau CC, Williams JG. The Peeling of Flexible Laminates. Int J Fract 1994; 66: 45−70. [22] Bueche F, Viscosity self-diffusion and allied effect in solid polymers, J Chem Phys 1959; 20: 1959– 1964.
Table 1. Predicted consistency index and power coefficient of the fitted dissipation function for the PE/G and PE/aFG interfaces. PE/G PE/aFG Temperature Consistency Power Consistency Power (K) index coefficient index coefficient 220 1270.7 0.0616 1451.6 0.0598 230 1257.2 0.0622 1406.6 0.0600 240 935.4 0.0935 1362.5 0.0687 250 843.7 0.0998 1107.8 0.0758 260 737.6 0.1176 1027.8 0.0881 270 763.9 0.1055 937.4 0.0926 280 465.5 0.1754 844.6 0.1050 290 385.7 0.1790 719.3 0.1334 300 312.9 0.2010 630.6 0.1484 310 185.3 0.2653 525.3 0.1647 320 128.4 0.2853 426.4 0.1775 Table 2. Predicted energy dissipation function for the PE/G and PE/aFG interface. Interface Dissipation function RS* ) )
PE/G
PE/aFG
)
)
)
)
)
)
)
)
)
)
)
0.993
)
*RS is R-square value. Figure 1. Temperature dependence density of the PE model. Figure 2. Evolution of the orientation autocorrelation function of end-to-end distance unit vector,, for PE chains at PE/G and PE/aFG simulated interface. 24
0.994
Figure 3. Snapshots of stretching simulation of PE/G interface at: (a) initial step and (b) 70 Å displacement, and (c) for PE/aFG at 60 Å displacement (G single sheet, PE chains, and NH2 groups are represented in black, gray and blue, respectively). Figure 4. (a) Stress-displacement and (b) stress-strain behavior of the PE/G interface at different stretching velocity and temperature of 300 K. Figure 5. The snapshot of deformation process of the PE/G interface at the different stretching rates just after interface failure: (a) 0.25, (b) 0.5, (c) 0.75 and (d) 1 m/s (G single sheet and PE chains, are represented in black and gray, respectively). Figure 6. Interfacial fracture dissipation function, , as a function of starching rare at different temperatures. Figure 7. Stress-displacement response of the PE/G interface at the different temperatures and stretching rate of 0.25 m/s. Figure 8. The effect of amino functionalization of the G surface on the interfacial fracture dissipation function (). Figure 9. Density profile of the PE chains on the G and aFG surfaces. Figure 10. Master curves of the fracture energy dissipation function for: (a) PE/G and (b) PE/aFG interfaces. (Shift factors vs. temperature curve are shown inside the figure, which were applied for both the interface master curves)
Figure 1 25
Figure 2
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Figure 3
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Figure 4
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Figure 5
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Figure 6
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Figure 7
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Figure 8
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Figure 9
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Figure 10
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PII: DOI: Reference:
S0143-7496(18)30071-X https://doi.org/10.1016/j.ijadhadh.2018.02.032 JAAD2157
To appear in: International Journal of Adhesion and Adhesives Received date: 3 October 2017 Accepted date: 16 February 2018 Cite this article as: Sousa Javan Nikkhah, Mohammad Reza Moghbeli and Seyed Majid Hashemianzadeh, A Quantitative Correlation between Polyethylene/Graphene Interfacial Viscoelastic Dissipation and Deformation Parameters: A Molecular Simulation Study, International Journal of Adhesion and Adhesives, https://doi.org/10.1016/j.ijadhadh.2018.02.032 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A Quantitative Correlation between Polyethylene/Graphene Interfacial Viscoelastic Dissipation and Deformation Parameters: A Molecular Simulation Study Sousa Javan Nikkhah1, Mohammad Reza Moghbeli 1*, Seyed Majid Hashemianzadeh2 1
Smart Polymers and Nanocomposites Research Group, School of Chemical Engineering, Iran
University of Science and Technology (IUST), Tehran 16846–13114, Iran 2
Molecular Simulation Research Lab, Department of Chemistry, Iran University of Science and
Technology (IUST), Tehran 16846–13114, Iran
Correspondence to: Tel: +98 21 77240519; Fax:+98 21 77240495.
[email protected] Abstract In this study, the influence of applied tensile stretching rate and temperature on amorphous polyethylene/graphene (PE/G) and polyethylene/amino functionalized graphene (PE/aFG) interfaces were studied via molecular dynamics (MD) simulations. The simulations results indicated that the interfacial adhesion during stretching process was increased with increasing the stretching rate (V) while decreased with increasing the temperature (T). This behavior can be connected to changing the polymer chains friction arisen from inadequate reorientation time, which especially happened at the higher stretching rates and lower temperatures. Furthermore, the amino functionalization of G strengthened both the thermodynamic work of adhesion and interfacial viscoelastic energy dissipation function, (V, T), in the interface deformation process. In fact, relying the chains on aFG surface with stronger interfacial interactions also increased the required energy for the chains to disentangle and reorient on the surface. Moreover, the stretching rate dependency of the value well follows a power law at all the stretching temperatures. The effect of stretching rate and temperature on value was simultaneously
1
investigated utilizing the time-temperature superposition principle. Master curves of the simulation data, (aTV), was obtained for PE/G and PE/aFG interfaces on which the horizontal (aT) and vertical shift factor (bT) were calculated using Williams-Landel-Ferry (WLF) equation and Bueche-Rouse theory, respectively. Keywords: Dissipation function, Interfacial fracture, Master curve, Polyethylene, Graphene, MD simulation. 1. Introduction Various carbon nanostructures such as graphene exhibit exceptional mechanical, thermal and physical properties with ability to considerably reinforce polymer matrix in their nanocomposites. The mechanical properties of reinforced polymers with carbon nanoparticles are influenced by interfacial adhesion between nanofillers and polymer matrix due to an effective load transfer at the interface. The structural and dynamic properties of polymer chains at the interface are extremely changed because of the interfacial confinement effects. Polymer chains in nanocomposites have a large interface area with nanoparticles; hence, the interfacial properties has a crucial role on the ultimate mechanical properties of nanocomposites, which arises from the load transferring from the polymer matrix to the nano-filler through the interfacial region in between. Therefore, it is essential to predict the mechanical behavior of interfacial region at different temperatures and stretching rates to tune its performance during deformation [1]. Amico et al. [2] and Huang and Wang [3] showed that the viscoelastic dissipation energy originated from adhesive fracture and friction at polymer/filler (substrate) bond, depends on the thermodynamic work of adhesion (W), applied bond detachment rate and temperature in both the bulk and interface regions.
2
The adhesion fracture energy, G, is defined as the energy per unit area required to separate two surfaces from each other. The fracture energy is consisted of the thermodynamic work of adhesion (W) and rheological viscoelastic dissipation function depending on the bond detachment rate (V) and temperature (T) [4, 5]. Maugis and Barquins [5] proposed the G value according to the following equation: ))
(1)
where W is the thermodynamic work of adhesion, φ is the dissipation function, and αT is the William–Landel–Ferry shift factor for time-temperature superposition principle [6]. Gent and Schultz [4] declared that dissipation energy, which is required for fracture of a viscoelastic adhesive bond from an adherent, is remarkably larger than the W. Kadiyala et al. [7] investigated the adhesion fracture energy and failure mechanism of thermoplastic polymer/metal interfaces via lap shear test at high temperatures. The G dependency for different polymer/solid adhesive bonds have been suggested based on experimental results. Maugis et al. [5] performed the peel test of polyurethane layer adhered to a glass substrate. Their results showed that the dissipation energy as a function of crack velocity followed a power-law model with a power-law coefficient of 0.6. Although, the strength of interfacial region can remarkably influence the mechanical properties of nanocomposites and adhesive bonds, it is impossible to characterize and quantitatively study only the polymer/solid interface region using current laboratory equipments. Additionally, different simultaneous molecular fracture mechanisms cannot be easily distinguished from each other during interface deformation. This is where molecular simulation techniques, such as molecular dynamic (MD) simulations, have been considered as valuable tools to visualize a deep insight of molecular orientation and layering at the interface region [8]. Clear dependency of
3
simulated interfacial adhesion on the detachment rate and temperature in a molecular scale would explain the interface chain responses to the rheological variables during bond deformation and provide the ability to compare the dissipation energy at the interfacial and bulk regions. MD simulations prepare detailed information about the chemical or physical phenomena occurred at the interface in nanoscale dimensions [1, 8]. The simulations can promote the explanations of experimental data, and open a direction toward tuning the interface strength [8]. In nanocomposites, a strong polymer/nanofiller interfacial area could considerably influence the rheological and mechanical properties, although it is too difficult to determine the role of interface and bulk regions separately. MD simulation would predict the properties of interfacial region separately, which cannot be estimated using experimental tools [8,9]. Some researchers tried to investigate the failure of polymer/substrate interface via MD simulations. However, few works were carried out to investigate the failure behavior of thermoplastic polymer/substrate interfaces [1, 8‒10]. Yuan et al. [1] investigated the interfacial normal strength of three typical polymers, i.e. polyethylene, polyurethane and polystyrene, confined between two graphene single sheets via normal stretching MD simulations. In our previous work, adhesion strength between PE and graphene was calculated using tensile MD simulations [10]. The main goal was to study the effect of polymer chain length, number of polymer chains, and graphene surface functionalization on the fracture behavior and adhesion strength at the interface. In fact, the interface was studied from the molecular ordering, configuration, and energetic point of views during its stretching simulation. For this purpose, the evolution of density profile of the polymer chains at the interface region and the evolution of different potential energy contributions during deformation process were evaluated. Additionally, ordering and configuration were studied via quantifying order parameter and chain disentanglement during the stretching process,
4
respectively. However, there are other important parameters such as stretching rate and temperature, which would influence the interfacial adhesion strength due to the viscoelastic behavior of the polymer chains at the interface region. Therefore, it seems a comprehensive study of the interface from the viscoelastic point of view is necessary to deepen insight about adhesion mechanisms at the interface region during its deformation. In addition, the quantitative evaluation of the interface viscoelastic energy dissipation as a function of above-mentioned rheological parameters can determine interface capability to resist against failure or debonding at different deformation conditions. Despite some published simulation works to determine polymer/nanoparticle bond strength [1, 8‒ 10], little attention has been paid to present a quantitative mathematical correlation between the fracture energy and bond detachment variables only in an interfacial region. In the present research work, the interfacial adhesion between polyethylene (PE) and graphene (G) or amiofunctionalized graphene (aFG) has been studied using MD simulation at various stretching rates and temperatures. The goal of this research work is to investigate the interfacial fracture behavior and the dependency of the viscoelastic dissipation energy function (φ) of the interface region to the bond loading parameters, stretching rate and temperature.
2. Simulation details 2.1. Models construction In the present study, PE as a commercial, easily processable and low-cost thermoplastic was selected as a polymer matrix, which its interfacial adhesion on a single sheet graphene was examined. The amorphous cell procedure [8, 10], has been used to produce the atomistic PE model.
energy of PE box was minimized based on the following procedure. The box
5
was subjected to an initial energy minimization using Conjugate Gradient algorithm. Thereafter, the polymer was subjected to a NPT (constant particle number, pressure and temperature) simulation at 300 K for 10 ns at the pressure of 0.0001 GPa after a NVT (constant particle number, volume and temperature) equilibration at 300 K for 8 ns. The first NVT simulation was necessary for pre-equilibration and elimination of energy barriers. Afterward, the bulk density of the pure PE and complete equilibration of the simulation box were obtained during the final NPT simulation at atmospheric pressure. The final equilibrated periodic PE box containing 40 PE chains each one with 300 carbon atoms and dimensions of 68.20×68.20×68.20 Å3, exhibits a density value of 0.88 g/cm3, which is in a good agreement with experimental data published pr
[11]. In order to ensure that an
equilibrated periodic model of PE chains had been obtained, glass transition temperature (Tg) of the model in addition to its density was evaluated. The Tg value of a PE model can be calculated according to instant change in the slope of its density-temperature curve. The protocol of Tg calculation was started by the cooling process of the PE model. At the beginning, the polymer model was equilibrated at 600 K. Afterwards, the temperature was decreased from 600 to 50 K during 22 steps. At each step, a 2000 ps NPT simulation was performed for equilibration. In this case, the data sample was a set of data selected from the last 200 ps of the simulation running. The density variation versus temperature of the PE model is illustrated in Figure 1. The approximate Tg value is considered as the temperature once an obvious instant change in the slope of density-temperature curve has been observed. The calculated Tg value of 220 K is in a good agreement to experimental Tg values reported in the range of 190–300 K by some researchers [9]. As expected, the calculated Tg values of PE model should be lower than those of experimental values because of their lower molecular weights in comparison to commercial PE
6
samples. Nonetheless, much faster cooling rate in MD simulations in comparison with real applied cooling rates in experiments caused the predicted Tg values be increased [10]. The agreement between the calculated physical properties of PE model, density and Tg values, and experimental results verifies achieving to the equilibrated PE model. A parallel MD code LAMMPS [12] was applied to perform the equilibration and the following simulations. Sun et al. [13–15] for several organic and in-
validated classes. Nose-Hoover thermostat
and Berendsen barostat were used to control the temperature and the pressure, respectively. The Verlet velocity algorithm was applied for integration with 1fs time step. The non-bonded cutoff distance of 12 Å was used. pristine graphene (G) and the amino-functionalized graphene (aFG) with 4.06 NH2 functional groups per nm2 were used as two nano-fillers and adherent substrates. The G model consists of 1824 carbon atoms with the total surface area of 68.66×66.74 Å2. The selection of the aforementioned amino-functional surface density was due to the highest interfacial adhesion obtained with the PE chains, when compared to the other surface densities [8]. In the case of aFG, the amino-functional groups were assumed to randomly attach onto only one surface of the neat graphene by chemical covalent bonding. The functionalization . An energy minimization run of the neat graphene and aFG was performed using a Conjugate Gradient algorithm. To create PE/G and PE/aFG interfaces, G and aFG surface were introduced near the surface of the polymer at a distance less than the cutoff radius, respectively. The simulation interface boxes were further enlarged to 2000 Å in the z direction to assure that the adjoining systems did not
7
interact in the z direction, when 3D periodic boundary conditions were applied. First, the joint systems were heated up to 600 K for 1000 ps and then equilibrated at 300K for further 18000 ps using NVT ensemble. The average amounts of results for the last 1000 ps were considered to calculate the energy of the systems. The height of the polymer layer on the G or aFG interface at the equilibrium state was about 68.23 Å. The autocorrelation function of end-to-end unit vector,
the top rigid region
along the z-direction [10] The separation 0.5 Å i
8
Å step of stretching, the equilibration process was performed for different duration times to achieve different stretching rates. Additionally, the stretching process in each rate was carried out at different temperatures in order to study the effect of both temperature and stretching rate on the fracture energy at the PE/G or PE/aFG interface. Since the applied forcefield in the simulation could not predict the bond breakage, it was assumed that the chains would not break while slip out in a viscous process. The normal stress was calculated applying the Virial theorem [16] all through the only middle moveable region. The stress in each step of deformation was determined by time-average over the latter half of the equilibrium interval and was recorded versus displacement to exhibit the stress-displacement behavior of each interface system.
3. Results and discussions 3.1. Thermodynamic work of adhesion (W) The functionalization of graphene surface was carried out to investigate the effect of different interfacial interactions on the interfacial fracture energy when compared with the neat graphene. The W term, the reversible free energy change for creation of two new equilibrated free surfaces, which quantifies the thermodynamically interfacial adhesion between two surfaces, was evaluated for the PE/G and PE/aFG interfaces according to the following equation [17,18]. ))
(2)
where A and EInt are the contact area and interaction energy between the PE and G or aFG surface, respectively. EG-PE is the total potential energy of the equilibrated PE/G or PE/aFG interface, EG is the potential energy of the G or aFG, and EPE is the potential energy of the neat polymer. According to the simulation results, the calculated W values of the PE/G and PE/aFG interfaces were equal to 175.93±7.23 and 825.55±12.03 mJm-2, respectively. Accordingly, the 9
attachment of amino groups on the G surface seems to considerably improve the dispersive forces and, therefore, interfacial interactions between the PE and aFG surface. Determining the role of W on the fracture energy (G) of the PE/G and PE/aFG interfaces brings about a deep insight about the surface treatment of reinforcement materials. Although strongly depends on the stretching rate and temperature [4, 5], the proper dissipative process can be revealed if the viscoelastic adhesive layer adheres tightly on a substrate with strong interfacial interactions. However, there is a question about the quantitative role of W in the fracture energy at the interfacial region. Furthermore, the stretching rate and temperature dependency of the , only at the interface region, is also another important interest in designing nanocomposite materials. 3.2. Tensile simulation results Figure 3 shows the stretching process of the PE chains in the PE/G and PE/aFG interfaces. In fact, the interface deformation was arisen from the consequent steps of the chain stretching process. The stress-displacement and stress-strain results obtained from tensile simulation of PE/G interface at a given temperature, i.e. 300 K, are indicated in Figure 4. In order to calculate the strain, the displacement was divided by the initial length of the un-rigid part of the polymer layer in each stretching step. As shown, the stress-displacement curve is almost linear at the beginning of the process before yielding is occurred. After the yield point in the post yielding region, the stress starts to decrease because of the PE chains disentanglement. As shown in Figure 4, the yield stress increases with increasing the stretching rate. This increment can be attributed to increasing friction between the polymer chains, which causes more resistance against the deformation. At the lower stretching rates, the role of post-yielding region is prompted because the chains have more time to adopt with each deformation step. In fact, the chains are able to slide, rotate and disentangle to exhibit more viscous behavior. From
10
thermodynamic point of view, it can be suggested that for tensile process at high stretching rate, high monomer friction makes constraints at different points throughout the chain. Therefore, lowering the numbers of configurations (Ω), which can be experienced by the chains, reduces the entropy (S) of the system [19]. This reduction originated from system ordering, which makes it difficult for the chains to relax after each stretching step and increases the resistance against chains deformation at high stretching rates. In contrast at low stretching rates, enough time for chains adoption reduces the friction and, therefore, increases Ω. This behavior enhances the chains disordering and increases the S value. Consequently, the chains at each stretching step have opportunity to experience further rotation and slipping in order to relax and reduce their resistance against the applied deformation. Figure 5 shows the snapshots of deformation process for the PE/G interface with different stretching rates just after interface failure. At lower stretching rate, the deformation seems to occur almost throughout the PE layer (Figs. 5a‒ b). On the contrary, the PE chains in high stretching rate separated from the graphene surface. As shown, only some chain segments could deform and the other segments just near the graphene surface remained undeformable (Figs. 5c‒ d). These different behaviors can be related to the above thermodynamic discussion about different chain resistance versus deformation at different stretching rates. The interfacial interactions for the PE/G interface at different stretching rates are similar. However, the applied stress at higher stretching rate could not easily deform the beneath layers of PE due to more friction among the chains and higher resultant resistance against the deformation. Therefore, the stress just transferred throughout PE layer until it reached to PE/G interface contact region. Consequently, an adhesive failure was observed at the interface before the chains were deformed completely. On the other hand, a fairly cohesive failure mode was
11
observed at the lower stretching rates because of more accessible time for chains deformation. The applied force at the low stretching rates was able to deform the different layers of PE. Depending on strength of the interface and deformation rate, the failure may take place either at the interface region in contact to graphene surface or in the bulk region. In this case, the PE/G interface experienced different fracture modes based on stretching rate applied in the deformation process. 3.3. Interfacial fracture energy Experimentally, measuring the interfacial fracture energy between polymer and a solid surface especially at micro- and nanoscale is a challenging subject because the polymers are viscoelastic and soft when compared to rigid inorganic solid materials. In the present study, the interface fracture energy was calculated based on the area underneath the stress-displacement curve. The simulation stress-displacement results for the PE/G interface at different stretching rates are shown in Figure 4. The stress-displacement curve at the stretching rate of 0.25 m s-1 exhibits a typical elastic region which is sequentially followed by a yield, disentanglement, and separation region. The elastic modulus, the initial slope of linear part of the curve, and the yield stress increased with increasing the stretching rate. At the higher stretching rate, the disentanglement and separation region are also observed but not dominant as well. One of the primary aims of general line of this research has been to find a mathematic relationship between the viscoelastic dissipation function (φ) at the interface region and debonding variables, i.e. stretching rate and temperature. 3.3.1. Effect of stretching rate The viscoelastic dissipation function can be calculated in terms of G and W according to Eq. 1, that is, φ = (G-W)/W. The G value of the interface at a given temperature and stretching rate
12
was calculated from the area underneath of its corresponding stress-displacement curve and W calculated using Eq. 2. Figure 6 indicates the effect of stretching rate on the viscoelastic dissipation function at different temperatures around the glass-transition temperature (Tg) of the PE. As shown, the value increases with increasing the stretching rate in all temperatures. This observation can be attributed to the predominant viscous response at lower stretching rates. At lower stretching rates, the chains have more time to equilibrate and relax after each separation step, and then slip on each other in a liquid-like behavior. This observation can also be described by Deborah's number,
, the ratio of chain stress relaxation time (τr) to observation time
(τp), which characterizes the fluidity of materials. De depends either τr or temperature change of material. For pressure sensitive adhesive De is defined as an equation,
, where V and l
are stretching rate and initial adhesive thickness, respectively. The De value lowers with decreasing V, indicating more viscous behavior of interface over a substrate at the lower stretching rates [20]. However, increasing the stretching rate improved the value due to predominant elastic response which reflected the higher required stress for chain orientation. Increasing the stretching rate increases De, meaning the elastic behavior is the dominant response. In addition, the friction against chains motions and slippage increases because of short available time for chains reorientation (τp). These elastic response and friction at higher stretching rates resulted in higher interfacial fracture energy. Based on the thermodynamic point of view, the free energy of the system in canonical ensemble, i.e. Helmholtz free energy (A), can be defined according to two approaches of state thermodynamic function and partition function (Q) given by the following equations: (3) 13
where U, S and T are internal energy, entropy and absolute temperature, respectively. In addition (4) in which Q can be obtained as follows: ∬
[-
)
]
(5)
where p, q, N and H are momenta, position, number of particles and total energy (sum of kinetic energy K(pN) and potential energy V(rN)), respectively [19]. Since increasing the stretching rate reduces S, the change in A value at constant temperature can be determined by the U value with changing the stretching rate. Increasing the stretching rate at a constant temperature and given same displacement can increase U value due to more system deviation from its equilibrium state. The amount of A increases with increasing the stretching rate. On the other hand, increasing the stretching rate causes the system experiences the states with higher total energies, H (pN, rN), during deformation process. Therefore, the A value increases with increasing the stretching rate because the system undergoes the states with higher total energy throughout the deformation process. The abrupt reduction of value at the very high stretching rates and higher temperatures occurred during the deformation process (Figure 6). It should be mentioned that decreasing at the very high stretching rates above 240 K can be attributed to insufficient time for chain reorientation after each stretching step. Therefore, it would not be possible for the PE chains to adopt well with each new deformation steps. At this condition, the chains would not be able to slide, stretch, rotate and disentangle. In fact, the chains behave more rigidly and cannot endure the applied stress which can decrease the value. Consequently, the simultaneous presence of strong interfacial interactions, chain disentanglement capability and chains friction during deformation of an interface could result in a high value. 14
Figure 6 shows the variation versus V on a log-log scale plot at the temperatures below 320 K. Fitting power-law function to the simulation data at the various temperatures resulted in the coefficient values in the rage of 0.0616‒ 0.2853. The consistency index and power coefficient values of the power-law relations fitted to the simulation results at different temperatures are summarized in Table 1. As shown, the dissipation function has not significantly changed with increasing the stretching rate at the temperatures near the Tg of PE. It is suggested that at these temperatures, the PE still behaves glassy enough without showing significant viscous energy loss during the tensile deformation. The obtained consistency index indicating the elastic behavior (stiffness) of the interface and power-law coefficients are almost similar at these lower temperatures. In contrast, the value considerably increased with increasing the stretching rate at the higher temperatures above 240 K due to the deformation occurring inside the PE chains during the stretching process. The previous research works showed that the viscoelastic dissipation energy of an adhesive bond significantly increased with increasing of stretching rate. The trend of the presented simulation results in a molecular scale is in a good agreement with experimental results published by Gent and Schultz [4] and Kinloch et al. [21]. It should be mentioned that unlike the simulation method, in experimental tests both the polymer bulk layer and interfacial region contribute in the amount of the . In this research work, the MD simulation provides the capability to investigate the deformation behavior and to calculate the only in the PE/G interfacial region. In addition, the type of the considered adhesive in the present work and the experiments are different. Therefore, the difference between the calculated exponent coefficient values of the simulation and those of experimental values is reasonable. Nonetheless, a similar trend in variation of the 15
interfacial fracture energy as a power-law function in both the simulation and experimental research works was observed. 3.3.2. Effect of temperature The debonding temperature influences the chain stretching through varying the thermal fluctuations and internal monomer friction. Figure 6 indicates the effect of temperature on the function of the PE/G interface. As expected, the value has decreased with increasing the temperatures. By increasing the temperature at a given stretching rate, more chains with high thermal energy are accessible. The temperature raising increases Q and lowers the amount of A, which means a lower G value. Figure 7 shows the stress-displacement curves for the PE/G interface stretched at stretching rate of 0.25 m/s and temperatures of 220, 270 and 320 K. The temperature changing exhibited an obvious effect on the stress-displacement behavior of the interface. As expected, the material stiffness reduces with increasing the temperature. The stress-displacement responses at 220, 270 and 320 K show a similar behavior, i.e. an elastic region and a yield point followed by a gradual displacement. However, it is observed that the maximum stress and fracture energy at 320 K is much smaller than those at 220 and 270 K. This implies that the interface strength becomes weaker with temperature increase in the interface. It was assumed that the chains would not break during debonding while could slip out through a viscous process. Additionally, increasing the temperature decreased the consistency index while enhanced the power coefficient. This observation indicates the reduction of interface elastic behavior with more viscous flow by temperature increasing. In addition, De reduces by increasing the temperature due to reducing the τr value. In this state, the polymer interface behaves more viscously.
16
3.3.3. Effect of graphene surface functionalization Figure 8 indicates the stretching velocity dependency of for PE/aFG interface at the various temperatures. As shown, an increase in value was observed with functionalizing the graphene surface at the given temperatures. The attachment of amino groups on the graphene surface increased W [8, 10] due to stronger dispersive interactions. Furthermore, the functionalization also influenced the rheological dissipation behavior in the interface (Figure 6 and 8). This behavior can be attributed to more intensive layering effect of the polymer chains on the aFG when compared to the graphene surface. Figure 9 shows the chain reduced density, i.e., the interface chain density to bulk chain density ratio (ρ/ρ0) as a function of the distance from G or aFG) surface. For both the PE/G and PE/aFG interfaces, a sharp density profile peak was observed at the vicinity of the G or aFG surface, which indicated a dense segment layer near the surface and layering behavior of the PE at both the interfaces. The intensities of other peaks were reduced gradually as the distance from the surface increased. As shown, the intensities of the peaks in the PE/aFG interface are higher than those of the PE/G interface, which can be related to the stronger interfacial interaction at the PE/aFG interface. It can be understood that functionalization of the graphene at the interface led to creating more ordered chain layering region in comparison to the neat PE/G interface. Additionally, the functionalization with amino groups increased the required energy to disentangle and deform PE chains in this well ordered region during the elongation process. Furthermore, stronger interfacial interactions cause more constraints for PE chains at the PE/aFG interface. A less numbers of configurations (Ω) and, consequently, a lower S value was experienced by the chains on the aFG surface when compared to graphene surface at the same temperature, stretching rate and displacement. It is more difficult for the chains to relax at the PE/aFG system after each deformation step due to more tight
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binding of the chains to the aFG surface. Therefore, the U and A values of the PE/aFG interface are higher than those of the PE/G one due to more deviation from equilibrium state. The consistency index and power coefficient amount of the fitted power-law functions to the calculated values for PE/aFG interface at each stretching process temperature are summarized in Table 1. As shown, the consistency index values of the functionalized interface is larger than those of the unfunctionalized interface. Stronger interactions between the chains and amino groups on the graphene surface increased the fracture dissipative energy. On the contrary, the power-law coefficient for the PE/aFG interface in a given temperature is smaller than that of the PE/G interface. This means that the change rate of value with the stretching rate at the functionalized interface is slower than that of the un-functionalized interface. In fact, the chemically bonded amino groups on the graphene surface improve the interfacial interactions and adsorb the chains more tightly than the neat graphene surface. Accordingly, it can be suggested that stronger interactions at the PE/aFG interface cause the chains can rely on the aFG surface and then more chains volume can be deformed before interface failure at the higher stretching rates. 3.3.4. Time-temperature superposition Time-temperature superposition is a valuable method to explain the viscoelastic behavior of polymers in a wide range of time and temperature via making a master curve on which the obtained data in different temperatures are shifted on a reference temperature [6]. The effect of stretching rate (time) and temperature on the energy dissipation function was simultaneously investigated utilizing time-temperature superposition principle. The horizontal shift factor values were calculated according to Williams-Landel-Ferry (WLF) equation [6]. )
(6) 18
where αT is the William–Landel–Ferry shift factor and T is the test temperature. The WLF equation is usually used when the Tg is selected as the reference temperature and T is in the range of Tg
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amount of V and decreasing T value. At higher log αT value, more stress is required to disentangle the chains at the interface. 4. Conclusions In the present study, the correlation between the energy dissipation function of the PE/G and PE/afG interfaces and stretching rate and temperature was investigated. Accordingly, at the low temperatures, the interfacial dissipation function did not significantly change with increasing the stretching rate. At the lower temperatures, the PE interface behaved glassier than the high temperatures in the deformation process. On the contrary, at the higher temperatures above the Tg value, a significant change in the dissipation function was observed with increasing the stretching rate due to the higher viscoelastic dissipation associated with the plastic deformation occurred inside the interface. Fitting power law to the simulation data resulted in the power-law coefficient in the range of 0.0616‒0.2853 for the PE/G interface, and in the range of 0.0598‒ 0.1775 for the PE/afG interface. At the given temperature, the power-law coefficient for the PE/aFG interface was smaller than that of the PE/G interface. This means that the variation of the PE/aFG interface was slower than the PE/G interface. The stronger interfacial interactions at the PE/aFG interface provides the possibility for the chains to slip out in a more viscous process, especially at the higher stretching rate. In fact, the higher interfacial fracture energy is achievable when chains exhibit both elastic and viscous behavior during the stretching process. This behavior can be observed for interfaces with stronger interactions, which enable the adhesive chains to disentangle and reorient properly. Therefore, the resultant friction arisen from these chains movements causes the interface to resist further against the deformation. The present research
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work can provide the opportunity to study the interfacial region from the thermodynamic and rheological point of views and design an interface with higher strength for proper load transfer. Although this present research work has been devoted to investigate the interface between an amorphous PE and graphene, it is obvious that the presence of crystalline phase in the polymer structure at the interface can influence on the interfacial adhesion. Therefore, the chains deformation, sliding and rotation may take place with difficulty. This issue and considering semi-crystalline instead of amorphous PE in simulations will be our future research work. Acknowledgment The authors gratefully appreciate the Iranian Nanotechnology Initiative Council and the visepresident for research and technology of Iran University of Science and Technology (IUST) for their partial financial supports.
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References [1] Yuan Z, Lu Z, Yang Z, Sun J, Xie F. A Criterion for the Normal Properties of Graphene/Polymer Interface. Comput Mater Sci 2016; 120:13−20. [2] Amico FD, Carbone G, Foglia MM, Galietti U. Moving Cracks in Viscoelastic Materials: Temperature and Energy-Release-Rate Measurements. Eng Fract Mech 2013; 98:315−325. [3] Huang YH, Wang J. Prediction of Coating Adhesion Loss due to Stretching. Int J Adhes Adhes 2013; 40: 49−55. [4] Gent AN, Schultz J. Effect of Wetting Liquids on the Strength of Adhesion of Viscoelastic Material. J Adhes 1972; 3: 281−294. [5] Maugis D, Barquins M. Fracture Mechanics and the Adherence of Viscoelastic Bodies. J Phys D: Appl Phys 1978; 1:1989−2023. [6] Williams ML, Landel RF, Ferry JD. Mechanical Properties of Substances of High Molecular Weight .19. The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-Forming Liquids. J Am Chem Soc 1955; 77: 3701−3707. [7] Kadiyala AK, Bijw J. Investigations on Performance and Failure Mechanisms of High Temperature Thermoplastic Polymers as Adhesives. Int J Adhes Adhes 2016; 70:90−101. [8] Nikkhah SJ, Moghbeli MR, Hashemianzadeh SM. Investigation of the Interface between Polyethylene and Functionalized Graphene: A Computer Simulation Study. Curr Appl Phys 2015; 15:1188−1199. [9] Awasthi AP, Lagoudas DC, Hammerand DC. Modeling of Graphene–Polymer Interfacial Mechanical Behavior Using Molecular Dynamics, Modelling Simul Mater Sci Eng 2009; 17: 015002.
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[10] Nikkhah SJ, Moghbeli MR, Hashemianzadeh SM. Dynamic Study of Deformation and Adhesion of an Amorphous Polyethylene/Graphene Interface: A Simulation Study. Macromol Theory Simul 2016; 25: 533−549. [11] Brandrup J, Immergut EHE, Grulke A. Polymer Handbook. New York: Wiley-Blackwell; 1999. [12] Plimpton SJ. Fast Parallel Algorithms for Short-Range Molecular Dynamics, J Comput Phys 1995; 117: 1−19. [13] Sun H. Force Field for Computation of Conformational Energies, Structures, and Vibrational Frequencies of Aromatic Polyesters, J Comput Chem 1994; 15: 752. [14] Sun H, Mumby SJ, Maple JR, Hagler AT. An ab Initio CFF93 All-Atom Force Field for Polycarbonates, J Am Chem Soc 1994; 116: 2978−2987. [15] Sun H. COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase Applications: Overview with Details on Alkane and Benzene Compounds. J Phys Chem B 1998; 102: 7338− 7364. [16] Tsai DH. The virial Theorem and Stress Calculation in Molecular Dynamics. J Chem Phys 1979; 70:1375. [17] Kisin S, Vukic JB, van der Varst PG, With G, Koning CE. Estimating the Polymer-Metal Work of Adhesion from Molecular Dynamics Simulations. Chem Mater 2007; 19:903‒ 907. [18] Henry DJ, Yiapanis G, Evans E, Yarovsky I. Adhesion between Graphite and Modified Polyester Surfaces: A Theoretical Study. J Phys Chem B 2005; 109: 17224‒17231. [19] Widom B. Statistical Mechanics, A Concise Introduction for Chemists. New York: Cambridge University Press; 2002. [20] Reiner M. The Deborah Number. Physics Today 1964; 17: 62.
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[21] Kinloch AJ, Lau CC, Williams JG. The Peeling of Flexible Laminates. Int J Fract 1994; 66: 45−70. [22] Bueche F, Viscosity self-diffusion and allied effect in solid polymers, J Chem Phys 1959; 20: 1959– 1964.
Table 1. Predicted consistency index and power coefficient of the fitted dissipation function for the PE/G and PE/aFG interfaces. PE/G PE/aFG Temperature Consistency Power Consistency Power (K) index coefficient index coefficient 220 1270.7 0.0616 1451.6 0.0598 230 1257.2 0.0622 1406.6 0.0600 240 935.4 0.0935 1362.5 0.0687 250 843.7 0.0998 1107.8 0.0758 260 737.6 0.1176 1027.8 0.0881 270 763.9 0.1055 937.4 0.0926 280 465.5 0.1754 844.6 0.1050 290 385.7 0.1790 719.3 0.1334 300 312.9 0.2010 630.6 0.1484 310 185.3 0.2653 525.3 0.1647 320 128.4 0.2853 426.4 0.1775 Table 2. Predicted energy dissipation function for the PE/G and PE/aFG interface. Interface Dissipation function RS* ) )
PE/G
PE/aFG
)
)
)
)
)
)
)
)
)
)
)
0.993
)
*RS is R-square value. Figure 1. Temperature dependence density of the PE model. Figure 2. Evolution of the orientation autocorrelation function of end-to-end distance unit vector,
0.994
Figure 3. Snapshots of stretching simulation of PE/G interface at: (a) initial step and (b) 70 Å displacement, and (c) for PE/aFG at 60 Å displacement (G single sheet, PE chains, and NH2 groups are represented in black, gray and blue, respectively). Figure 4. (a) Stress-displacement and (b) stress-strain behavior of the PE/G interface at different stretching velocity and temperature of 300 K. Figure 5. The snapshot of deformation process of the PE/G interface at the different stretching rates just after interface failure: (a) 0.25, (b) 0.5, (c) 0.75 and (d) 1 m/s (G single sheet and PE chains, are represented in black and gray, respectively). Figure 6. Interfacial fracture dissipation function, , as a function of starching rare at different temperatures. Figure 7. Stress-displacement response of the PE/G interface at the different temperatures and stretching rate of 0.25 m/s. Figure 8. The effect of amino functionalization of the G surface on the interfacial fracture dissipation function (). Figure 9. Density profile of the PE chains on the G and aFG surfaces. Figure 10. Master curves of the fracture energy dissipation function for: (a) PE/G and (b) PE/aFG interfaces. (Shift factors vs. temperature curve are shown inside the figure, which were applied for both the interface master curves)
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Figure 2
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Figure 5
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Figure 6
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Figure 7
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Figure 8
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Figure 9
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Figure 10
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