Anastassia N.
Rissanou
*a,
Petra
Bačová
a and
Vagelis
Harmandaris
*ab
aInstitute of Applied and Computational Mathematics (IACM), Foundation for Research and Technology Hellas (FORTH), GR-71110 Heraklion, Greece. E-mail: risanou@uoc.gr; Fax: +30 2810393701; Tel: +30 2810393746
bDepartment of Mathematics and Applied Mathematics, University of Crete, GR-71409, Heraklion, Crete, Greece. E-mail: harman@uoc.gr; Fax: +30 2810393701; Tel: +30 2810393735
First published on 11th July 2019
The dynamical behavior of nanographene sheets dispersed in polymer matrices is investigated through united-atom molecular dynamics simulations. The Brownian motion of the sheet and the anisotropy in its translational and orientational diffusion are the topics of the current study. Different polymer matrices and pristine and functionalized graphene constitute various nanocomposite systems. Interactions between the nanographene flake and the matrix determine the dynamics of the systems. The dynamics is reduced in polyethylene oxide compared to polyethylene matrix, whereas carboxylated sheets move considerably slower than the pristine nanographene in any matrix. Diffusion is anisotropic for short times, while it becomes isotropic in the long time limit. The in-plane motion of the nanographene sheet is faster than the out-of-plane component, in agreement with the diffusion of perfectly oblate ellipsoids. In functionalized graphene, the anisotropy is suppressed. By exploring the temperature effect on both the nanographene sheet and polymer close to the surface, indications for coupling in the motion of the two components are revealed. The strong effect of edge functional groups on the dynamics can be used as a way to control the Brownian motion of nanographene sheets in polymer nanocomposites and consequently tailor the properties of the materials.
In the current study the Brownian motion of nanographene flakes in polymer matrices is investigated. Graphene is a two-dimensional monolayer of graphite of macroscopic dimensions but of atomic thickness, which was first isolated in 2004.11 Nowadays, there is enormous interest in the study of graphene-based nanostructured materials, due to the exceptional physical properties of graphene (e.g., electronic, optical, thermal, and mechanical properties),11–15 which make it a candidate material for a wide range of potential applications.16–20 In polymer nanocomposites graphene sheets are used as nanofillers21–25 in order to enhance their mechanical and functional properties.
The conformational and dynamical properties of nanographene in a polymer matrix can be of particular importance. Indeed, both the conformational transitions (rippling or wrinkling) and the mobility of the graphene nanofiller in the composite can strongly affect the properties of the whole material;26–30 however, there is no obvious correlation between them. For example, it has been observed that both these factors inhibit adsorption of polymer on the surface of a graphene flake in polyethylene/graphene nanocomposites, resulting in a lower density of polymer at the interface, compared to the case of a frozen sheet.31,32 Furthermore, the strong effect on the electronic properties of graphene30,33 and increased chemical reactivity and change in charge distribution34,35 have been attributed to rippling.
Rubinstein and coworkers36,37 have explored the way that a polymer melt affects the motion of a spherical nanoparticle through scaling theory and molecular dynamics simulations. This is found to be related to the correlation between the diameter of the nanoparticle (d) and the structural length scales of polymers (i.e., spacings between polymer entanglements) (a). The mobility of the nanoparticle is enhanced for d < a, remains almost unchanged for d ≈ a and is suppressed for d > a.
For a 2D flake, anisotropic Brownian particle dynamics has been detected experimentally through optical trapping of individual graphene flakes in water.38 The study of the Brownian motion of ellipsoidal particles in water reveals an anisotropic diffusion for short times, which becomes isotropic for longer times. The coupling between the rotational and translational motion of the ellipsoids was illustrated with the use of digital video microscopy.39 Furthermore, the motion of perfectly rigid prolate ellipsoids dispersed in a sea of spheres has been investigated through molecular dynamics simulations using simple bead–spring models.40 It was found that there is anisotropy in the motion of the ellipsoids for short up to intermediate times, where the motion of the molecules is faster in the direction parallel to their long (major) axis compared to the perpendicular direction, characterized as needlelike motion. An opposite behavior is observed for oblate ellipsoids, where the correlation between the two components of the mean squared displacement is reverse (i.e., faster perpendicular and slower parallel components with respect to the major axis).41 In both cases for long times the diffusion becomes isotropic.
In the present work we study the (anisotropic) Brownian motion of small graphene sheets (nanographene) in different environments. Since there is no accurate theoretical description of the dynamics of a fluctuating 2D material in a polymeric (viscoelastic) medium, we have used predictions from simpler models, concerning the diffusion of non-spherical objects in a Newtonian liquid, and more specifically of oblate ellipsoids; nanographene flakes are expected to be, approximately, described by such a shape.
The current study highlights the dynamics of nanographene flakes well-dispersed in polymer matrices. Both different polymers and different kinds of nanofillers (i.e., pristine and functionalized graphene sheets of various sizes) have been examined. In general, this is a complicated issue since the dynamics is expected to depend on several factors, such as the size and the shape of the sheet, the functional groups at edges, and the strength of the graphene/polymer interactions. In all cases model graphene flakes are of rather small areas (nm dimensions); therefore, the term nanographene is used. The goal of the present study is to reveal information about the (anisotropic) dynamical behavior of nanographene on the atomic scale. The effects of different host matrices, types of nanographene sheets and temperatures on the dynamics of the sheet are presented. The translational and rotational motions of nanographene are examined in both parallel and perpendicular directions to the graphene plane. The anisotropic dynamical behavior of the nanographene sheet, induced by its 2D geometry and its conformational transitions, are also detected. Moreover, a coupling between the motions of the nanographene flake and the polymer layer in its vicinity is investigated.
The rest of the paper is organized as follows: in Section 2 we briefly discuss the basic theoretical concepts of anisotropic Brownian motion. In Section 3 details about the model systems, the simulations, and the analysis procedure are given. The Results and discussion section separated into subsections according to the individual parameters under investigation are presented in Section 4. Conclusions follow in Section 5.
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There is a big controversy about the solutions of the Navier–Stokes hydrodynamic equations depending on the boundary conditions which are applied. According to the stick hydrodynamic boundary conditions, D‖ is two times faster than D⊥ for rods and the predictions are almost identical for a prolate ellipsoid, for the aspect ratios (κ) greater than 2 (κ = L/b, with L the length and b the diameter of the rod). The use of slip hydrodynamic boundary conditions predicts that the motion along the parallel direction can be decoupled from that along the perpendicular direction and the D‖/D⊥ ratio approaches κ for large κ-values. This applies to both prolate and oblate ellipsoids.41
Vasanthi et al.40,41 presented results of molecular dynamics simulations, using bead–spring models, which show a linear variation of D‖/D⊥ with κ, supporting the slip hydrodynamic boundary conditions. According to their findings, anisotropy is observed for long times, in the case of oblates with the motion normal to the major axis (D⊥) being much faster than the one parallel to the major axis (D‖). The trend was opposite for prolate ellipsoids and anisotropy was shown for much shorter time scales.
Nanographene's shape is approximately described by an oblate ellipsoid, as is briefly described in the ESI,† (Table SI-1). Therefore, keeping the convention of Fig. 1c, the major axis corresponds to the Z′-direction. Then the corresponding notation in our calculations is −X′Y′ for motion normal to the major axis and −Z′ for motion parallel to the major axis.
If we also consider the short time regime, then an effective time dependent diffusion coefficient, D(t), can be defined as:
For an anisotropic particle, following the discussion in the previous subsection, we can further define the components of the time dependent diffusion coefficient as and
, where (ΔR2)Z′ and (ΔR2)X′Y′ are the components of the mean squared displacement for motion parallel and normal to the major axis, respectively. For the motion normal to the major axis the X′ and Y′ directions are equivalent, so their average is used.
United-atom models are used for both polymers, whereas an all-atom description is used for graphene flakes. For PE and PEO the TraPPE force field45 has been used, which has been slightly modified in the case of PEO.46 TraPPE has been widely used in the literature for both PE and PEO simulations,31,32,47 leading to reliable results that compare fairly well with existing experimental data.48 For PEO 1620 10-mer polymer chains and for PE 1336 22-mer polymer chains were simulated. The sizes (end-to-end vectors) of both PEO and PE chains are about 18–20 Å, and their radii of gyration are about ∼6.5 Å at 450 K. Concerning the model graphene sheets, we applied a force field previously used for various carbon structures,49,50 which has been developed through ab initio calculations. For the functional groups grafted onto the graphene edges, since there are no available quantum data, we have chosen the OPLS-AA force field.51 OPLS-AA is a quite general force field widely used in the description of charged groups. Nanographene sheets of three different dimensions have been simulated in order to estimate the size effect on the various properties under investigation. In all cases the sheet is almost quadratic. A detailed description of the systems, as well as of the atomistic force field, used to describe interactions for both polymer matrices and graphene flakes, is given in the ESI.† All simulated systems are presented in Table 1 and a characteristic configuration of a model nanographene/polymer nanocomposite is depicted in Fig. 1a. The acronyms of our systems are as follows: G stands for graphene, H/COOH subscripts denote the hydrogenated/carboxylated flakes, respectively, and the type of polymer matrix follows after ‘/’. In the case of pristine graphene the subscripts of (G) denote the different areas of the sheet.
Acronym | Atoms of graphene | System | T (K) | Simulated time (ns) |
---|---|---|---|---|
Pristine graphene/PE | ||||
G20/PE | 190 | Graphene (19 × 20) Å2/PE | 450 | 100 |
G50/PE | 1032 | Graphene (49 × 51) Å2/PE | 450 | 100 |
G80/PE | 2546 | Graphene (84 × 86) Å2/PE | 450 | 100 |
Functionalized graphene (49 × 51) Å2 | ||||
GH/PE | 1122 | Graphene-H/PE | 450 | 100 |
GCOOH/PE | 1302 | Graphene-COOH/PE | 450 | 200 |
GH/PEO | 1122 | Graphene-H/PEO | 450 | 200 |
GCOOH/PEO | 1302 | Graphene-COOH/PEO | 450 | 300 |
(GCOOH/PEO)400 | 1302 | Graphene-COOH/PEO | 400 | 300 |
(GCOOH/PEO)370 | 1302 | Graphene-COOH/PEO | 370 | 300 |
(GCOOH/PEO)340 | 1302 | Graphene-COOH/PEO | 340 | 300 |
(GCOOH/PEO)318 | 1302 | Graphene-COOH/PEO | 318 | 300 |
Molecular dynamics simulations were performed in the isothermal–isobaric (NPT) statistical ensemble at a constant pressure of 1 atm. The integration time step was either 1 fs or 2 fs, depending on the system (i.e., systems with functionalized graphene sheets are simulated with a smaller time step, due to the high vibrational frequency of bonds between edge groups and graphene atoms). All bonds were constrained using the LINCS algorithm.52 All the systems were first equilibrated and then followed by production runs, the time of which varies for the different systems between 100 ns and 300 ns as shown in Table 1. Configurations were saved every 50 ps. The temperature for the PE/graphene systems was kept constant using the stochastic velocity rescaling algorithm,53 while in the case of the PEO/graphene ones the Nosé–Hoover thermostat was used.54 The time constant for temperature coupling was 0.2 ps for the stochastic velocity rescaling thermostat and 2.5 ps for the Nosé–Hoover thermostat. Correspondingly, the Berendsen barostat, with a time constant of 0.5 ps, and the Parrinello–Rahman barostat, with a time constant of 5 ps, were used. The Coulomb cutoff scheme was applied to the nonpolar matrix, and the particle-mesh Ewald (PME) electrostatics was present in PEO composites. For further details about the model and simulation procedure the reader is referred to the ESI† and our previous work.47 Finally, the GCOOH/PEO system was studied at different temperatures in the range of 318–450 K. Note also that for all the systems multiple (from 2 up to 4) simulations, starting with different uncorrelated initial configurations, were performed to improve statistics.
Translational Brownian motion (BM) is examined via the calculation of the mean squared displacement (MSD) of the nanographene sheet, in the body frame, as follows:
(a) Given a set of nanographene's snapshots, we define, for each configuration, an instantaneous reference coordinate system, X′Y′ and Z′, as discussed above.
(b) Then, the coordinates of all atoms of the rest of the trajectory are projected onto the instantaneous reference system.
(c) MSD X′Y′ and Z′ components are calculated with respect to the body frame of a given configuration for all subsequent times. The two components X′ and Y′ represent motion normal to the major axis (see Fig. 1c and the corresponding discussion), whereas the component Z′ describes motion parallel to the major axis.
Orientational motion is also analyzed in the body frame, by defining the three Eulerian angles which are formed in the body frame system of axes, between the initial and a given configuration. These angles express the direction cosines between two sets of axes, as is shown in Fig. 1b. Two angles describe the rotation of the graphene flake around an axis perpendicular to the graphene optimal plane (Z′) (rotation, θ1, θ2) and one angle represents the rotation around an axis parallel to the plane (X′ or Y′) (rotation, θ3).
In all the above cases, the multiple time origin technique has been used to improve statistics.
The factors that mainly affect the dynamics of the nanographene sheets in the nanocomposite are the size of the sheet (Table 1), the polymer/graphene interaction, and the different zero-shear viscosities of the two polymer matrices. In order to examine the latter, we provide a rough estimation for the zero-shear viscosity of the polymer matrix assuming Rouse dynamics. In more detail, the zero-shear rate viscosity of the polymer chains is computed via, where ρ is the mass density of the bulk system, M is the molecular weight of the polymer, R is the gas constant, T is the temperature, 〈Ree2〉 is the mean squared end-to-end distance of the chains in the bulk and D is the self-diffusion coefficient of the center of mass of the chain.32 The values obtained for n0 are 2.35 × 10−4 P and 8.70 × 10−3 P for PE and PEO respectively.
Fig. 2b depicts the mean squared displacement, scaled with 6t (time dependent effective diffusion coefficient), for the three nanocomposites of PE with pristine graphene flakes of different sizes (G20/PE, G50/PE, G80/PE). As expected, the smaller the sheet the faster its translational motion in the polymer matrix.31 In Fig. 2c (inset of Fig. 2b) the diffusion coefficient (D) is presented as a function of the size of the sheet. Values are extracted as a rough estimation of the plateau region of the curves. Not surprisingly, the dynamics of (single-sheet) well dispersed nano-graphene layers is rather noisy; however, a clear decrease of D with size is observed, which has a difference of almost one order of magnitude between the smallest sheet and the largest sheet.
Furthermore, from the data of Fig. 2 one can extract a characteristic time for the translational motion of the nanographene flakes. We define τtrans as the time during which a graphene sheet moves a distance equal to the half of its size, in one direction (e.g., 〈ΔR2〉 ∼ (25 Å)2 for G50). The corresponding values are also depicted in Table 2 and indicated with arrows in Fig. 2. Based on τtrans, it is interesting to observe that the graphene flakes in the PEO nanocomposites are almost 4 times slower for both hydrogenated and carboxylated sheets compared to those in the PE nanocomposites. Moreover, the carboxylated sheets are almost ∼2.5 times slower than the hydrogenated ones in both polymers.
System | τ trans (ns) ± ∼(20%) | τ orient (ns) ± ∼(20%) | β ± ∼0.05 |
---|---|---|---|
G50/PE | 3.5 | 8.92 | 0.83 |
GH/PE | 3.4 | 9.42 | 0.79 |
GCOOH/PE | 9.4 | 13.13 | 1 |
GH/PEO | 13.5 | 25.07 | 0.85 |
GCOOH/PEO | 31.5 | 48.96 | 1 |
GH/PE(80mer) | 10.5 | 25.32 | 1 |
Summing up at this point, we have to mention that although the comparisons of different graphene flakes in the same matrix are straightforward, comparing between different matrices is a multi-parameter task. The diffusion of graphene, in addition to the energetic interactions, depends on the molecular weight of the chain, the chain's dimensions, as well as the glass transition temperature and the viscosity of the polymer matrix. The above results are based on polymer chains of the same dimensions and similar molecular weights. For reasons of completeness an additional comparison was performed using 80mer PE chains which are of similar viscosity to 10mer PEO chains. The results for τtrans of the GH/PE (80mer) system are presented in Table 2. The dynamics of graphene flakes is reduced in the PE 80mer matrix compared to the PE 22mer matrix and τtrans is 10.5 ns, comparable to, but still shorter than, the one in the PEO 10mer.
We should state here that the time scales reported in the present work would not describe the BM of macroscopic graphene. Naturally, the larger the graphene sheets, the slower their dynamics, at both the translational and orientational levels. Therefore, for time scales as those considered here, macroscopic graphene would be expected to be still in the unsteady-state (time dependent) diffusion regime. However, a 30 ns time scale is enough for the nanographene sheets, studied in the current work, to reach the isotropic time independent (linear) regime.
The orientational dynamics of the nanographene sheets can be further quantified by defining a characteristic relaxation time for the half-diagonal vector, as the time integral of the P1(t) curves shown above. To achieve this, P1(t) data presented above are fitted with a KWW function , where A is a pre-exponential factor, which takes into account relaxation processes for very short times, τKWW is the relaxation time and β is the stretch exponent, which takes into account the deviation from the ideal Debye behavior. Then a characteristic orientational relaxation time is defined as the time integral of P1(t):
. Note here that fitting of these curves is restricted to times up to about ∼10–20 ns, beyond which data become rather noisy, especially for the systems of PEO nanocomposites. Results for the orientational relaxation time (τorient) and the exponent β are presented in Table 2. In the PEO matrix the hydrogenated graphene is ∼2.5 times slower than that in the PE matrix, while the carboxylated sheet is ∼4 times slower in PEO compared to PE. Moreover, the carboxylated flakes are slower than the hydrogenated ones ∼1.5 times in the PE matrix and ∼2 times in the PEO matrix. β-values show a rather narrow distribution of relaxation times for all the systems, as β is in the range 0.8–1 in all cases. The size dependence of the orientational dynamics is presented in the inset of Fig. 3 and has been analyzed in our previous work.31 As in the case of the translational dynamics, decorrelation is faster for the smaller sheets.
A similar comparison in the orientational motion of the functionalized graphene flake between the GH/PE (80mer) and GH/PEO systems has been performed and the corresponding results for τorient are presented in Table 2. Again the motion of the hydrogenated sheet in PE 80mer exhibits similar orientational dynamics to that of PEO.
Results for the MSD components of nanographene (pristine and functionalized) sheets in the PE and PEO matrices are presented in Fig. 4a and b. The anisotropic character of nanographene's Brownian motion is apparent in all the systems: there is a clear difference between the two components of MSD for short times (i.e., the subdiffusive regime), which, as expected, is eliminated for longer times, since the diffusion of the sheet in the long time regime becomes isotropic (i.e., linear regime). The motion normal to the major axis (X′Y′ component of MSD) is faster compared to the one parallel to the major axis (Z′ component). Furthermore, from the data shown in Fig. 4 it is clear that (a) for short times the difference between the X′Y′ and Z′ components of the mean squared displacement is higher for pristine graphene compared to the carboxylated one in the same matrix (PE) (Fig. 4a); (b) by comparing the two different matrices, a slightly smaller difference between the two components exists in the case of PEO, which underlines the effect of polymer–graphene interactions (Fig. 4b); and (c) in all cases the difference is gradually reduced and beyond a certain time, about ∼[20–30] ns, the diffusion becomes isotropic (i.e., linear regime – Fickian diffusion). The systems with hydrogenated graphene behave very similarly to the ones with pristine graphene.
Then we have calculated effective diffusion coefficients for graphene flakes, defined as and
and their D‖(t)/D⊥(t) ratio as a function of time is presented in Fig. 4c for the G50/PE, GCOOH/PE and GCOOH/PEO systems. D‖(t)/D⊥(t) attains the lowest value ∼0.26 for pristine graphene, but comparable values ∼0.5–0.6 for carboxylated graphene in the two polymer matrices, for short times. The difference between the two diffusion coefficients is bigger in the case of pristine graphene, where the two components of MSD have the highest deviation (Fig. 4a). Conformational transitions of the graphene flakes can be thought of as a possible reason for this observation. Indeed, we have observed bigger ripples in the pristine nanographene compared to the functionalized sheets, which result in bigger deviations from the ideal shape (i.e., oblate ellipsoid). Functionalization of graphene is an important factor which suppresses the rippling; therefore, it affects the dynamics. Representative results concerning the amplitudes of the ripples are presented in Table SI-3 (ESI†). Moreover, carboxyls are bulky groups, added to the edges, which are lying out-of-plane, hindering the in-plane motion. Therefore, they constitute an additional reason for the suppression of the anisotropy in the carboxylated graphene flakes. The D‖(t)/D⊥(t) ratio approaches one only in the limit of long times when the two components of the MSD almost coincide and the diffusion becomes isotropic. Moreover, the above calculations are in qualitative agreement with the predictions based on eqn (1)–(3) for oblate ellipsoids, since the D‖(t)/D⊥(t) ratio for nanographene sheets of (50 × 50) Å2 area is <1, in a range of [0.26–0.7], for the different systems. Values are constant for short times, when orientational decorrelation of the sheets has not been achieved yet, whereas they reach 1 later on. The value predicted from eqn (1)–(3) for a perfectly oblate ellipsoid of (50 × 50) Å2 area is 0.7 in good agreement with our calculations.
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Fig. 5 Time autocorrelation functions of 〈cos![]() |
Similar to pristine graphene is the behavior of the hydrogenated sheet in the PE matrix, whereas in the PEO matrix the difference between the two components is less pronounced (Fig. 5b). In Fig. 5c results for the carboxylated sheet in the two matrices are presented. For the GCOOH/PEO system autocorrelation functions attain still high values for the time window of the current simulation; therefore, we cannot extract any conclusion for their anisotropy. For the GCOOH/PE system the two components are identical for short times up to ∼10 ns, in the following the X′Y′ component is faster up to ∼30 ns before data become noisy.
Overall, interactions between the graphene flake and the matrix seem to be crucial for both the translational and orientational motions of the sheet. The stronger the interactions the lower the induced anisotropy, which is highlighted in the PEO nanocomposites. Furthermore important is the role of the end functional groups which also suppress anisotropy. Anisotropic diffusion can induce anisotropy in the overall properties of the nanocomposite, which in some cases is important to be controlled or at least understood.
The retardation of the translational dynamics at lower temperatures is obvious. On top of that it is interesting to observe that, in the time window of the simulations, the motion of the graphene flake is not diffusive in the PEO matrix at any temperature, but the highest one (450 K), where a plateau in Deff(t) is observed. A decrease in temperature leads to a broadening of the non-linear, subdiffusive regime of the BM of the nanographene sheet in a polymer matrix.
An analogous comparison among the different temperatures is performed for the orientational motion, through the calculation of the autocorrelation function of the first Legendre polynomial, P1(t), for the half diagonal vector of the graphene sheet (Fig. 1b). Results, which are presented in Fig. 6b, illustrate the retardation of the orientational motion with temperature as well. Characteristic times for the decorrelation of the vector are extracted from fits with KWW functions and are discussed below. Note here that at the lowest temperature value (318 K) the decorrelation of the half diagonal vector ACF of the sheet is limited in the time window of the simulation; thus the characteristic time that we have extracted from the KWW fit is used as a rough estimation of the slowdown of the graphene dynamics with decreasing temperature. For this reason a different (star) symbol is used in the corresponding graph (Fig. 8a).
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Fig. 7 The X′Y′ and Z′ components, defined on the body frame, of the mean squared displacement as a function of time for the GCOOH/PEO system at three different temperatures. |
Fits with an exponential function of the Arrhenius form (t = Aexp(−B/T)) are performed (dashed lines). For the polymer chains the decay of the relaxation time with temperature provides an exponent B equal to −4076.9 K for chains in the vicinity of the graphene layer, and −3092.8 K in the bulk. Similarly, an exponential decay is observed for the decorrelation time of the nanographene flake with temperature, which indicates a coupling in the orientational motion of the polymer and nanographene. The exponential fit in the case of nanographene was made on the last four points and an exponent almost two thirds of the one stands for the polymer, very close to the surface, was extracted (B = −2452.1 K). An estimation of activation energies can be extracted from these fits, resulting in Eact_polymer_1st
layer = −33.9 kJ K−1 mol−1, Eact_polymer_bulk = −25.7 kJ K−1 mol−1 and Eact_graphene = −20.4 kJ K−1 mol−1. This shows that the effect of the temperature is stronger on the polymer chains (i.e., ∼1.5 times faster decorrelation of the orientation of the chains in the first adsorption layer, with temperature, compared to the nanographene flake). The activation energies of the polymer in the bulk regime and nanographene are comparable, with the latter still being a bit higher. Thus polymer relaxation is more sensitive to the temperature change at any distance from the surface compared to the nanographene flake.
On top of that, the mean squared displacement of the center of mass of the polymer chains, which lie on the graphene surface (i.e., the first adsorption layer of 1.5 nm width), is calculated and compared to the mean squared displacement of the center of mass of the nanographene sheet (lab frame). To quantitatively study the different dynamical behavior of polymer chains around nanographene, the area around the sheet is divided into two regions (one parallel to the surface and one edge region) and the dynamics is calculated independently for the different regions.47 Results for the GCOOH/PEO system at T = 450 K are presented in Fig. 8b, where dashed lines indicate the polymer's motion parallel to the nanographene sheet region and solid lines for nanographene's motion. Both curves are fitted with a power law function, for the same time period, which can be assigned to the sub-diffusive regime (dotted lines). A similar slope (0.85 for graphene; 0.9 for the polymer) of the MSD curves is observed for both components, which indicates a coupling in their translational diffusion too. For longer times the diffusive regime is observed for both components (i.e., slope ∼1). For very short distances from the nanographene, the polymer is strongly adsorbed on the surface of the nanographene flake; therefore its motion is affected by the motion of the sheet.
Calculations in the lab frame provide the following results: translational dynamics are similar for pristine and hydrogenated nanographene in PE matrices. In PE nanocomposites nanographene diffuses more rapidly than that in PEO ones. This result is attributed to stronger polymer–graphene interactions, but in addition both the higher zero-shear viscosity of PEO compared to PE and the difference in their Tg values are determinant factors. The effect of the carboxyl edge groups is strong, since the carboxylated graphene sheets move more slowly than both the pristine and hydrogenated ones in both matrices. This highlights the effect of the electrostatic and possible H-bond interactions (for the PEO matrix) between the polymer and the functionalized nanographene sheet. On top of that, the mass difference between the pristine and functionalized nanographene flakes affects their dynamics correspondingly. Similar factors govern orientational dynamics. Correlations among the factors that determine the dynamics of the different nanocomposite systems provide important piece of information which can be transferred to systems with macroscopic graphene as well.
Since graphene is a two dimensional material, theory predicts inhomogeneity in its Brownian motion in terms of anisotropy between two directions, the in-plane motion (X′Y′) and the out-of-plane motion (Z′). However, theoretical relations are valid only for perfect shapes, like oblate or prolate ellipsoids, where constant diffusion coefficients are derived from the solution of the Navier–Stokes hydrodynamic equations. Diffusion is anisotropic for short times, while it turns into isotropic in the very long time limit, and for oblate ellipsoids the motion perpendicular to the major axis is faster than the one parallel to the major axis. According to a shape analysis procedure, unperturbed graphene sheets are closely described by oblate ellipsoids (thin discs) (see the ESI†). However, rippling of graphene, which is extensively affected by the interactions with the polymer matrix, induces changes in the shape of the sheets. These changes have a strong effect on the dynamics of the flake. Analysis of dynamics in the body frame reveals anisotropy between the two directions of motion (parallel and perpendicular to the major axis), with the motion perpendicular to the major axis being always faster than the one parallel to the major axis. Differences are more pronounced for pristine graphene. The effect of functional edge groups results in suppression of the differences between the two components of motion. Anisotropy is observed for times ∼20–30 ns beyond which the diffusion becomes isotropic (linear regime). Similar observations concerning the anisotropy stand for the rotational motion as well.
The temperature effect has been studied through the analysis of the autocorrelation function of the first Legendre polynomial for the half diagonal vector of graphene, which provides an effective activation energy. An analogous calculation for PEO very close to the surface (1st adsorption layer), based on the terminal relaxation times (i.e., end-to-end vector), shows that the decorrelation of the polymer orientation with temperature is ∼1.5 times faster than that of the nanographene. For both the polymer and nanographene, orientational relaxation times follow an exponential decay. In addition, in the translational motion, the mean squared displacements of the centers of mass of the polymer and nanographene versus time attain similar slopes. These observations indicate a coupling in the motion between the nanographene sheet and the amount of polymer very close to it.
In whole, interactions between the nanographene flake and the matrix have a strong effect on both the translational and orientational motion of the sheet. The stronger the interactions the lower the induced anisotropy. Proper functionalization of the graphene flakes specifies polymer–graphene interactions but determines the conformational transitions of the flakes as well; therefore, it could be used as potential control of the Brownian motion of the sheets.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp02074h |
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