Atomistic reconstruction of pyrocarbons nanostructure from HRTEM data Jean-Marc Leyssale1 *, Gérard Vignoles1, Raphaël Vitti2, Jean-Pierre Da Costa2, Christian Germain2, Patrick Weisbecker1, Roland Pellenq3 1
Laboratoire des Composites ThermoStructuraux, (UMR 5801 CNRS/CEA/Snecma Propulsion Solide/Université Bordeaux 1), 3 allée de La Boetie, 33600 Pessac (France) 2 IMS (UMR 5218 CNRS/Université Bordeaux 1/ENSEIRB/ENSCPB), 3 Centre Interdisciplinaire des Nanosciences de Marseille, (UPR 3118 CNRS), Campus de Luminy; Case 913; 13288 Marseille Cedex 9 (France) * Corresponding author. E-mail :
[email protected] Abstract. A new method for the generation of atomistic models of dense nanotextured carbons is presented. This method is based on the statistical analysis of HRTEM images and their three-dimensional extension through image synthesis under constraint. The resulting 3D images then serve as an external potential bringing the atoms to site preferentially on the dark areas during a conventional simulated annealing simulation. Application of this method to the case of two laminar pyrocarbons, differing in their degree of disorder, highlights the promising nature of this approach.
1. Introduction Atomistic reconstruction methods are nowadays well-established tools for linking experimental characterization data to the atomic scale structure of matter [DOVE_2002]. Most of them are based on the reproduction of orientation-averaged structural features like the pair distribution functions (PDF) so that these methods are usually very efficient for isotropic systems. However, when one has to deal with materials displaying a neatly anisotropic nanotexture, like most dense graphene-based carbons, the success of such approaches rely strongly on the initial guess of the atomic structure [ZETTERSTROM_2005]. This means, in crude terms, the failure of these methods for that class of materials. For instance, two PDF-based computer reconstructions of the same material show a drastically different nanotexture: one, quite isotropic, is an amorphous carbon with a dominating sp2 character [ACHARYA_1999] while the other one is a strongly anisotropic stack of faulted graphene sheets [SMITH_2004]. Nowadays, the nanotexture of turbostratic carbons, like for instance the pyrocarbon matrices present in many industrial C/C composites, is ususally described from High Resolution Transmission Electron Microscopy (HRTEM) lattice fringe images [OBERLIN_2002]. Although, the information provided by these images is mainly qualitative, recent developments in image analysis techniques allow a finer and finer description of the nanotexture to be drawn [GERMAIN_2005]. We propose here a new atomistic reconstruction strategy based on the information provided by such HRTEM images. The three steps of this method: (i) quantitative structural and statistical analysis of experimental images; (ii) synthesis of 3D “HRTEM-like” images under an orthotropy condition; and (iii) molecular dynamics annealing simulations using the synthetic 3D images as a complementary energy term to the interatomic potential, are described in section 2. Results of the application of this method in the case of two laminar pyrocarbons, differing in their degree of disorder, are presented in section 3.
2. Methods 2.1 Image analysis and synthesis HRTEM images are subjected to statistical analysis according to the method … Topic 11
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More details about analysis and synthesis of pyrocarbons HRTEM images can be found in the accompanying paper [ref_poster_jean-pierre]. 2.2 Atomistic reconstruction
Starting from the 3D synthetic images, cubic simulation boxes of compatible size (number of fringes multiplied by the suited interlayer spacing d002) are created and randomly filled by carbon atoms (no heteroatoms are considered in this work) to generate a system at the suited density. Typical pyrocarbon values of d002 and of the density, respectively 3.5 Å and 2.1 g/cm3, are adopted. A short Metropolis Monte Carlo simulation with a hard sphere model of 1.3 Å diameter is performed in order to ensure that no C-C distance is below that value. At that point random initial velocities corresponding to a temperature of 6000 K are assigned to the system and a realistic interatomic potential, the second generation REBO potential [BRENNER_2002] is switched on. The REBO energy is defined as N
[
V REBO = ∑∑ f c (rij ) VR (rij ) − bijVA (rij ) i =1 j >i
]
(1).
where N is the number of atoms, rij is the distance between atoms i and j, fc(r) is a switching function that goes smoothly from unity for r = 1.7 Å to zero for r = 2 Å, VR(r) and VA(r) are respectively the repulsive and attractive potentials and bij is an empirical bond order term, depending on both the environments of atoms i and j, and which properly takes into account the different bonding possibilities of hydrocarbons. This potential, which forces the system to adopt proper carbon bonding, is complemented by an external energy term VIMAGE to impose the nanotexture of the 3D image to the system. We chose here VIMAGE to be a simple linear function of the grey level : N
V IMAGE = ∑ kim I (ri )
(2).
i =1
where kim is a proportionality factor and I(ri) is the grey level (ranging from 0 to 1 for respectively black and white voxels) corresponding to the position ri of atom i. A simulated annealing procedure is then launched using molecular dynamics together with the stochastic thermostatting method of Andersen [ALLEN_TILDESLEY]. The system is cooled from 6000 to 0 K at a “slow” and constant rate of 5 K per picosecond by adjusting the target temperature of the thermostat at each timestep. Such a procedure mimic the dynamics of carbon system cooled from the liquid state in the presence of an external field (the 3D image potential) bringing the system to a given nanotexture. Tuning the relative importance of the two terms of the potential, by changing the value of kim allows to find a good balance between finding low energy structures and obtaining the suited nanotexture. We found that a value of kim = 2 eV, approximately one fourth of the sp2 carbon energy is appropriate. After completion of the annealing and in order to avoid any bias that could introduce the image potential, this term is switched off and the REBO potential is replaced by the AIREBO potential of Stuart [STUART_2002]. This potential is a modification of the REBO potential, incorporating in an adaptive manner the van der waals interactions occurring between non bonded atoms (interactions between different graphitic sheets). At that point the system is a relaxed for some picoseconds at a constant temperature of 300 K before calculation of properties.
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3. Results and Discussion We don’t discuss here the details of the image analysis and 3D image synthesis processes which can be found in the accompanying paper. Our starting points are thus the two 3D “HRTEM-like” images shown in Fig. 1. As can be seen on this figure, the two blocks show stacks, on the vertical direction, of two-dimensional fringes with varying lateral extension and undulation. More precisely, the fringes observed on Fig. 1a (left) appear to be less extended and more curved than those of Fig 1b (right). The nanotexture observed on Fig. 1a is rather typical of a low temperature “as prepared” carbon while the one on Fig. 1b resembles more the one of a heat treated, or “graphitized” pyrocarbon. In what follows the nanotextures displayed in Figs 1a and 1b well be respectively noted as Low Temperature (LT) and High Temperature (HT) in analogy with the temperature history of the classes of pyrocarbons they attempt to model.
a)
b)
Figure 1 : Synthetic 3D “HRTEM-like” images obtained using the method of … for two laminar pyrocarbons: a: Low Temperature pyrocarbon (left); b: High Temperature pyrocarbon (right). Noting that Figs 1a and 1b comprises respectively 15 and 14 fringes in the stacking direction, we have created cubic periodic simulation cells of side length 52.5 Å (15 x d002) and 49.0 Å (14 x d002) for respectively the LT and HT atomistic models (this amounts for systems of respectively 12398 and 15249 carbon atoms). The atomistic configurations obtained for the two materials at the end of the simulated annealing simulation (and after further relaxation with the AIREBO potential) are shown in Fig. 2. As can be seen on this figure, the two atomistic models show a well pronounced laminar nanotexture, typical of pyrocarbons. Carbon bonding in these two models is almost entirely (around 99 % ) of sp2 nature for both models, the remaining 1% of carbon atoms having a sp3 and some traces of sp hybridizations (two carbon atoms are considered bound if they are distant of less than 1.8 Å). In order to better characterize the materials, we performed an anylysis of ring statistics according to the “shortest path ring” (SPR) algorithm of Franzblau [FRANZBLAU_1991]. We found 2.74 SPR per atoms in the LT model and 2.77 SPR per atom in the HT model which is relatively close to the value three rings per atom in graphite. Moreover, a similar repartition of rings are found in the two materials with respectively 80 and 82 % of six-member rings (C6), 10 and 9 % of 5-member rings (C5) and 10 and 9 % of 7member rings (C7) for the LT and HT models. Although these fractions of non-hexagonal rings (20 and 18 % for respectively the LT and HT models) are relatively high, we expect that lowering the annealing rate will increase the fraction of C6 rings (simulations performed at a five times larger cooling rate have produced models with only 65 % C6 rings). Note that only sp2 atoms were considered in this analysis and that in both cases a small fraction of 8-member rings (around 0.5 %) were found.
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a)
b)
Figure 2 : Reconstructed atomistic models of Low Temperature (2a, left) and High Temperature (2b, right) laminar pyrocarbons. General C-C bonds are shown with thin sticks, C-C bonds between atoms belonging to “pure C6 fragments” (see the text for the definition) are displayed with thick coloured sticks. To further investigate the nanotextural quality of our reconstructed models, we performed a cluster analysis of pure according to the following procedure: (i) search all the present in the system; (ii) among the C6 rings, search all the “pure C6“ rings, namely those which are made of atoms belonging only to C6 rings; (iii) perform a cluster analysis to find all the graphitic fragments made of connected “pure C6“ rings (two “pure C6“ rings being connected if they have an atom in common). Graphitic fragments are displayed in Fig. 2 with thick coloured sticks. Also no quantitative results are provided in this paper, a simple visual inspection of Fig 2 shows that fragments belonging to the HT model seems to be a little more extended and also a lot flatter than those of the LT model. It is also interesting to observe that these “pure C6” fragments can be made of hundreds of rings (thousands of atoms) indicating a relatively high level of graphitization in some parts of the systems. Finally, we terminate this topological discussion by adding that some single fragments can present Y-junctions (see the orange and light blue fragments, middle-right side of Fig 2b) where two parts of the same fragment are stacked on each other. Going back to the very beginning of this work, we show in Fig. 3 HRTEM images of the LT (Fig. 3a) and HT (Fig. 3b) pyrocarbons taken as 2D-slices from the 3D images of respectively Figs. 1a and 1b. These images are to be compared with those simulated from the LT (Fig. 3c) and HT (Fig. 3d) atomistic models using the NCEMSS HRTEM simulation software of Kilaas [KILAAS]. Apart from differences in contrast, the similarity in terms of fringes undulations and junctions, between initial images (Figs a and b) and their corresponding simulated images (Figs c and d) is obvious and confirm that the atomistic models contain most of the nanotextural information present in the HRTEM images. It can even be seen that the method corrects the high frequency horizontal noise present in the LT synthetic HRTEM image (Fig. 3 a).
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a)
b )
c
d
Figure 3 : HRTEM images of pyrocarbons. a: 2D synthetic slice taken from the Low Temperature synthetic 3D image (Fig. 1a); b: same as (a) for the High Temperature 3D image (Fig. 1b); c: simulated HRTEM image obtained from the reconstructed atomistic model of the Low Temperature pyrocarbon (Fig. 2a); d: same as (c) for the High Temperature atomistic model (Fig. 2b).
As a last check up of our atomistic models, we plot on Fig. 4 the reduced Pair Distribution Functions (PDF) computed for the LT (straight line) and HT (circles) models. The reduced PDF or G(r) is defined as :
G(r ) = 4πρ0 rg (r )
(3).
where r is the interatomic distance, ρ0 is the the number density and g(r) is the atomic pair distribution function defining the normalized probability to find two carbon atoms distant of r :
g (r ) =
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∑∑δ (r − r ) N
i =1 j ≠i
(4).
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Although it is theoretically possible to measure the PDF of pyrocarbons, the difficulty to obtain these materials (usually deposited on fibrous performs) in bulk quantity, needed to perform the X-ray or Neutron scattering experiments, probably explains this function has never been published for such materials. Nevertheless, we also show for comparison purposes the experimental G(r) of a nanoporous disordered carbon (dashed line) [PETKOV_1999] and the computed G(r) of ideal hexagonal graphite (dotted line) on Fig. 4.
Figure 3 : Reduced pair distributions functions of the LT (straight line) and HT (circles) atomistic models. Displayed in dashed and dotted lines are respectively the experimental PDF of a nanoporous carbon [PETKOV_1999] and the computed PDF of ideal hexagonal graphite.
It can be seen on Fig. 4 that the two models have almost identical PDFs. This is not surprising as the PDF is an oriention-averaged measure of local order and, in the length range investigated, the two models have very similar structures. When compared to the nanoporous carbon and to graphite we see that the two models have the typical structure of a dense carbon, with all intra- and inter-layer peaks present. However, when compared for instance to the nanoporous carbon PDF measurements of Petkov et al. [PETKOV], we see that some peaks in the pyrocarbon PDFs are wider and less pronounced (see for instance the third neighbour peak at 2.9 Å or the peak at 6.6 Å). This can be due to the relatively high fraction (around 20 %) of non-hexagonal rings in our models while Petkov’s PDF has been modelled with atomistic configuration comprising zero [ACHARYA_1999] to 5 [SMITH_2004] % of such rings.
4. Conclusion In this paper we have proposed a new method, based on HRTEM data, for the atomistic reconstruction of nanotextured carbons. This method has been applied to two pyrocarbons, differing in their degree of disorder and has led to two relatively satisfactory models. Indeed the nanotextures of the two models, as well as the differences between them, are fairly compatible with the initial images. Moreover, on the chemical point of view, the atomistic models are typical of sp2 carbons and present well defined aromatic domains, well stacked on the orthotropy axis. However, at that point, the number of non-hexagonal rings (20 %) is slightly larger that what we would expect (lower than 10 %) for such a dense pre-graphitic carbon. The latter point is confirmed by the observation of the PDF, showing larger peaks than for instance the nanoporous carbon studied by Petkov et al. [PETKOV]. Nevertheless, we have reasons to believe that this “chemical quality” can be neatly improved by Topic 11
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lowering the rate of the simulated annealing process or by including in this process, together with the 3D image, a penalty term based on an experimental PDF, if available. Such an approach has been used by Jain et al. [JAIN_2006] in the simpler case of low density porous carbons.
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