jeanflop/post-ocr-correction huggingface dataset:
is known, this method of analyzing K fields is useful and allows us to seek effective transport properties. This also permits a process dependent analysis of the REV determination by varying H15 H2, H3. 1. INTRODUCTION. — On considered a transport process, within a heterogeneous medium, ruled by the equation: where K is a tensor, function of the point. The variations of this function are quasi-periodic at an scale 1. The applications that we envisage can be, among others: (i) K is a diffusion molecular tensor and |/ a concentration; (ii) K is a thermal conductivity tensor and |/ the
(ii) K is a thermal conductivity tensor and |/ the temperature; (iii) K is the permeability tensor of a heterogeneous porous medium and |/ the average intrinsic pressure of phase. At a scale r0, for which the medium can be considered as homogeneous, several methods, that we do not discuss here [1], permit to produce the equation describing <v|/>, average of [/] on the volume r0; one has: The coefficient of transfer K* is defined by: where b is, on a representative cell, solution of the following problem: Many techniques of resolution can be developed for solving the problem defining
of resolution can be developed for solving the problem defining the vector b and calculating the coefficient K*, especially in the case of periodic media. We are interested here in fields K obtained in a discrete way, by measurement on a grid of points covering the domain studied, or by image treatment. Thus, the field K can correspond to the thermal conductivity of a composite medium, in the classical sense of the mechanics of continuous media, obtained from the treatment of micrographs. No simple method exists to complement the usual morphological analysis tools to give indications on the principal directions
morphological analysis tools to give indications on the principal directions and the main values of the transfer tensor associated with the macroscopic process involving the entire field studied. It is about here an application making use of the change of scale classical microscopically to macroscopically. Other applications make use of an heterogeneity at the macroscopic level. It is important, in this type of applications, to have also information complementary to the morphological analysis concerning the properties of K*, permeability at large scale of the system. The evaluation of these properties makes use of a new change of scale macroscopically to
makes use of a new change of scale macroscopically to an even larger scale. By example, at the paragraph 4, we consider the field of permeability of a heterogeneous medium presenting variations spatial at a scale well above that of the pore. The local permeability corresponds, in fact, to a property defined by taking the average on an ensemble of pores or measured on a small volume including several pores. In view of these applications, it is interesting to have at our disposal simple logics permitting to exploit immediately the fields of values thus obtained. We propose to apply to
fields of values thus obtained. We propose to apply to the resolution of equations (4) and (5) a method of the type analysis of Fourier which has the advantage of treating simply the conditions of periodicity and which is close to the usual techniques of signal treatment. We present here the big lines of the algorithm and the results of several examples of application. 2. METHOD OF RESOLUTION. — The idea is to use the method of Galerkin with decompositions of the type: where, by definition, and where the domain studied has for dimensions 2H15 2H2, 2H3. By multiplying (4)
studied has for dimensions 2H15 2H2, 2H3. By multiplying (4) by fmu ra2 m3 and integrating on the domain studied, one obtains finally the linear system following defining the components of the vector b: From the resolution of the linear system and of the equation (3) one obtains: Compte tenu du fait qu'un such algorithm produces full matrices, the possibility of resolution in 3D leads to truncate the series defining b at 15 x 15 x 15. 3. PROBLEM TEST. — Among other geometric configurations simple that we have studied (lattice of spheres, lattice of cylinders, ...), we present here
(lattice of spheres, lattice of cylinders, ...), we present here the results obtained in the case of a lattice of spheres. The thermal conductivity of the lattice of spheres has been the object of numerous studies [2]. The existing knowledge on this problem being very developed, it does not s'agit here only to verify the method proposed on a known case. The previous studies all agree to recognize the representativeness of Maxwell's formula for non adhesive lattices of spheres. The equivalent conductivity of the lattice is given by: where Xj and X2 are the conductivities respectively of the spheres and
and X2 are the conductivities respectively of the spheres and of the continuous phase, and E1 is the volumetric fraction occupied by the spheres. The figure 1 represents the evolution of the relative error of our results compared to this formula as a function of the contrast of the conductivities respectively of each phase. The characteristics of the lattice studied are defined on the figure. Obviously, the relative error increases with the contrast of conductivity but in a reasonable manner considering the number of terms in the approximation. 4. EXAMPLE OF APPLICATION: PERMEABILITY OF A GRANULAR MEDIUM. — It is
OF APPLICATION: PERMEABILITY OF A GRANULAR MEDIUM. — It is here a real case which illustrates the interest of our approach. Many works have been devoted to the permeability of real media (see references in [3]). One distinguishes two problems: (i) the determination of the principal directions associated with the permeability tensor K, (ii) the determination of the values of its components. In the principal reference, there exist methods allowing to provide bounds for the values of the components [4], but we do not know of simple methods for defining the principal directions. The algorithm proposed allows to solve both
the principal directions. The algorithm proposed allows to solve both these problems. [1] BENSOUSSAN A., LUCIANO, D., STROUD, I., GIRARD, G. and LADOUCE, R., 'Méthodes statistiques en physique : de la théorie aux applications', Masson, 1984. [2] CLERC M. and SADOUN G., 'Effective conductivity of two-component composites with arbitrary shapes', J. Physique, 47, 1986, p. 551-562. [3] SNETHEN R., 'Modelling flow in heterogeneous media with application to aquifers', in D. SCOTT, I. STROUD and F. WOODBINE (eds.), 'Homogenization techniques in the analysis of heterogeneous media', Hemisphere, New-York, 1989, p. 35-55. [4] TOUCHEMAN P., 'Bounds on the effective properties of composites',
[4] TOUCHEMAN P., 'Bounds on the effective properties of composites', in D. SCOTT, I. STROUD and F. WOODBINE (eds.), 'Homogenization techniques in the analysis of heterogeneous media', Hemisphere, New-York, 1989, p. 91-97. Sur la figure 2, est représenté le champ de perméabilité local mesuré sur un grès des Vosges [5]. Le milieu est supposé localement isotrope. Le champ K(x) mesuré est analysé par FFT (dans cette application particulière, nous avons disposé de 15x21 mesures), ce qui fournit les coefficients du développement (6). Le vecteur b calculé permet la détermination du tenseur K*; les vecteurs principaux de ce tenseur sont représentés
tenseur K*; les vecteurs principaux de ce tenseur sont représentés sur la figure 2. Les valeurs calculées des composantes principales sont respectivement 10,6 et 23,4 mD. 5. Un tel résultat illustre une méthode d'analyse qui permet d'associer à un champ de données, des propriétés équivalentes pour un processus de type diffusif en traitant convenablement les effets d'anisotropie. On peut, en principe, en modifiant la taille du champ analysé, étudier les longueurs de corrélation (VER Volume Élémentaire Représentatif, etc.) associées à ce processus diffusif. Note reçue le 23 juin 1988, acceptée le 6 octobre 1988. Fig. 2. — Champ de perméabilité
le 6 octobre 1988. Fig. 2. — Champ de perméabilité locale et perméabilité équivalente pour un échantillon de grès (courbes d'isoperméabilité en millidarcys). Fig. 2. — Local permeability field and large scale permeability for a sandstone sample (isopermeability curves in millidarcys). C. R. Acad. Sci. Paris, t. 307, Série II, p. 1687-1692, 1988 RÉFÉRENCES BIBLIOGRAPHIQUES [1] A. BOURGEAT, M. QUINTARD et S. WHITAKER, C. R. Acad. Sci. Paris, 306, série II, 1988, p. 463-466. [2] G. K. BATCHELOR, F. R. S. et R. O'BRIEN, Proc. R. Soc. Lond., A, 355, 1977, p. 313-333. [3] M. QUINTARD et S. WHITAKER, J.
1977, p. 313-333. [3] M. QUINTARD et S. WHITAKER, J. Mécan. Théo, et Appliquée, 6, n° 5, 1987, p. 691-726. [4] G. MATHERON, M. BEUCHER, C. DE FAUQUET, A. GALLI, D. GUÉRILLOT et C. RAVENNE, Society of Petroleum Engineers, 1987, 16753. [5] M. DUPUY, Y. MORINEAU et P. SIMANDOUX, Rapport, Institut français du Pétrole (A.R.T.F.P.), 1964, n° 10951. Laboratoire Énergétique et Phénomènes de Transfert, U.A. C.N.R.S. n° 873, Ecole nationale supérieure des Arts et Métiers, 33405 Talence Cedex. Physique de la matière condensée/Condensed Matter Physics RMN de 13C dans une sphère de pyrographite orienté Pierre LAUGINIE, Henriette ESTRADE-SZWARCKOPF, Bernard ROUSSEAU
sphère de pyrographite orienté Pierre LAUGINIE, Henriette ESTRADE-SZWARCKOPF, Bernard ROUSSEAU et Jacques CONARD Résumé — L'utilisation d'un échantillon de pyrographite orienté (H.O.P.G.) de forme sphérique a permis de déterminer avec précision l'anisotropie des déplacements de 13C dans un plan de graphène. Le résultat est comparé aux informations provenant de la RMN de 13C dans les composés d'intercalation du graphite et dans les molécules aromatiques. 13C NMR in an oriented pyrographite sphere Abstract — The use of a spherical "highly oriented pyrographite" allowed the 13C shift anisotropy of a graphène plane to be determined. Results are compared with 13C NMR in
to be determined. Results are compared with 13C NMR in graphite intercalation compounds and in aromatic molecules. Abridged English Version — In graphite intercalation compounds, it has been possible to observe the 13C NMR line due to each carbon layer and for the inner ones (stage >2), two kinds of carbon sites were distinguished. On the contrary, in graphite, until now, those two kinds of carbon sites, resulting from the peculiar A-B stacking of the graphène layers, had never been put into evidence. The reason for non-observation of these sites resides in the great magnetic susceptibility χm of graphite when
resides in the great magnetic susceptibility χm of graphite when the magnetic field H0 is applied parallel to the layer axis: χ= -48 x 10-6 whereas χx= -(0.7-1) x 10-6 emu.cm-3. In the usually shaped NMR samples, i.e. thin rectangular slabs of H.O.P.G. (Highly Oriented Pyrographite), the NMR observation is only possible in regions in which the R.F. field may penetrate. Due to strong conductivity anisotropy, those regions are as drawn on Figure 1a. Unfortunately, these regions are in a very inhomogeneous demagnetizing field when H0||c: this field may be distributed over 300 ppm. The only solution appeared to be
distributed over 300 ppm. The only solution appeared to be a sample in which the demagnetizing field would be homogeneous, for instance a sphere. In 1984, we tried such an experiment but unfortunately it failed: the Ho||c line was very broad because of the poor quality of the pristine graphite, chosen because it is easy to machine [6]. However, convinced that this was the solution [9], we tried again with a good H.O.P.G.. Using a device described on Figure 1b we obtained at last the spectra shown on Figure 2. While we were writing this Note, a paper from our
While we were writing this Note, a paper from our japanese colleagues came out [7] reporting a more complete but similar experiment on an ellipsoidally shaped H.O.P.G. All apparent discrepancies between their results and ours are easily explained: the differences in shifts are accounted for by the different magnetizing factors; our lines are broader because of a small axial hole in our sphere and our components of the doublet are apparently not equal because of too short pulse repetition rate used (100 sec.). Thus we have now all what is necessary to compare graphite with its intercalation compounds. If we
necessary to compare graphite with its intercalation compounds. If we consider a flat cavity containing an isolated single graphène layer inside the sphere, the local field undergone by the layer C-atoms may be expressed as: Note présentée par Jacques FRIEDEL. 0249-6305/88/03071693 $ 2.00 © Académie des Sciences C. R., 1988, 2e Semestre (T. 307) Série II 1694 C. R. Acad. Sci. Paris, t. 307, Série II, p. 1693-1697, 1988 where Ho is the applied field, Hlocal is the effective field measured by an NMR line position; Hmacroscopic is the sum of the demagnetizing and Lorentz fields of the whole sphere
of the demagnetizing and Lorentz fields of the whole sphere given by the usual formula: H, is the "Internal" field produced by the graphène plane itself. Similarly the anisotropy of the shift may be written as follows, where we replace by 5 all the relative fields values: In this formula we see that, in graphite, the greatest term comes from χ|| which is a lot smaller in GICs. Using the numerical data, we obtain for the interesting term 5^ - 5iJ in graphite: After correction due to the small dipolar interaction, we obtain -169 + 4 ppm for the sole
interaction, we obtain -169 + 4 ppm for the sole orbital contribution to the shift. This value is very close to that observed in frozen benzene [2] (—180 ppm) which is known to be essentially due to a paramagnetic shift in the HQ -L c direction. Moreover, in alkali GICs, the orbital term extrapolated to infinitely dilute compounds (zero density of states at EF) gives also a close value: —190 ppm ([3], [4]). So, thanks to this new NMR experiment, we may confirm the similar origin of the orbital anisotropy in graphite and GICs and in aromatic molecules. 1. WHY?
in graphite and GICs and in aromatic molecules. 1. WHY? — 1.1. The NMR of 13C in graphite intercalation compounds gives two types of sites within the graphene planes, "internal"; studies of displacements as a function of stage — therefore of doping — have allowed us to separate the contribution of the spin magnetism of the carriers (proportional to the density of states at the Fermi level) from that of orbital magnetism ([1] to [4]). This latter, little sensitive to the nature and to the concentration of the intercalant, presents an anisotropy of the order of 190 ppm [3]. A
an anisotropy of the order of 190 ppm [3]. A comparison with pure graphite is called for. 1.2. But the NMR in pure graphite encounters great difficulties when the field is applied parallel to the c axis: broad lines, non-reproducibility from one sample to another ([2], [4], [5]). In particular, the two types of atoms "A" and "B" (corresponding to the presence or absence of neighbors in the same position within the first neighboring planes) were never differentiated. We have attributed the origin of these difficulties [6] to the inhomogeneity of the so-called "demagnetizing" (in reality "remagnetizing") fields associated with
of the so-called "demagnetizing" (in reality "remagnetizing") fields associated with the very great diamagnetic susceptibility of graphite in this orientation: X|| = −48.10−6 μm.cm3, so that the variations of displacement linked to the external shape of the sample may extend over 47rxn = −602 ppm. Given the geometry of the fields and our frequency of measurement, the penetration of the radiofrequency into a parallelepipedic sample is indeed essentially through the faces (a, c) where the thickness of the skin attains 1 to 3 mm, whereas it is only 30 μm for the faces (a, b): the demagnetizing field is then
for the faces (a, b): the demagnetizing field is then very inhomogeneous within the irradiated regions (fig. 1 a). From here our decision to remedy by fabricating a sphere from HOPG. C. R. Acad. Sci. Paris, t. 307, Série II, p. 1693-1697, 1988 1695 2. HOW? — 2.1. Because of the difficulties of machining a material as easily machinable as HOPG, our sphere is constituted of two hemispheres glued to the ends of a tube of pyrex of the same diameter; a hole of diameter 2 mm, necessary to the machining, has been closed by two plugs of HOPG in
machining, has been closed by two plugs of HOPG in order to forbid the entry of the wave; the inhomogeneity of the field due to the remainder of the hole (length 6 mm) contributes notably to the linewidth but cannot modify the mean displacement by more than a few parts per million (fig. 1 b). 2.2. We have used a spectrometer with pulses and Fourier transform, working at 90.5 MHz within a field of 8.45 T. The progress compared to the parallelepipedic samples is spectacular: — a doublet, of separation 37 ppm, is observed when the field H0 is parallel
37 ppm, is observed when the field H0 is parallel to the c axis; — the line is unique and fine when H0 ⊥ c; — the anisotropy is established for a sphere to: 8|| − 8X = −563 ppm (average of the doublet). where the displacements are counted positively within the sense of increasing frequencies of resonance, and where the indices || and ⊥ refer to the orientations of the field parallel and perpendicular to c respectively. 2.3. We had already in 1984 attempted an experiment on a sphere cut from a pyrographite, relatively hard, and selected for its
cut from a pyrographite, relatively hard, and selected for its ease of machining. The spectrum obtained for H0||c was extremely broad (~400 ppm), of low intensity but its position was in agreement with the present results [6]. The distribution of the axes c does not explain, by itself, such an enlargement: the distribution of the local fields also depends on the mosaicity of the pyrographite. 2.4. During the preparation of this Note, an experiment very close, realized on an ellipsoid, is described by our Japanese colleagues [7]. The apparent discrepancy (66 ppm) between their results and ours is explained by
(66 ppm) between their results and ours is explained by the difference between the demagnetizing factors of the two samples. Furthermore, the two peaks of our doublet have unequal intensities. Fig. 1. — Arrangement of the HOPG samples and the NMR fields. Shaded areas: schematic view of the radiofrequency penetration, (a) Usual slab; (b) Sphere. 1696 C. R. Acad. Sci. Paris, t. 307, Série II, p. 1693-1697, 1988 This explains why the doublet is observed when the field is parallel to the c axis and not when it is perpendicular. The absence of a doublet in the latter orientation is
The absence of a doublet in the latter orientation is not, for the moment, explained. Our second motivation was to compare the anisotropy of the displacements of orbital origin within a graphene plane situated inside an intercalation compound with that of the same plane within pure graphite. The attention had already been drawn to the necessity of separating, not only the term of orbital origin but also the effects of the considerable magnetism of pure graphite which does not limit itself to the sole demagnetizing field ([6], [9]). Pour l'une ou l'autre des orientations principales d'un échantillon sphérique, le champ
ou l'autre des orientations principales d'un échantillon sphérique, le champ Hlocal, réellement vu par un noyau vérifie la relation : où Hj est le champ « interne » produit sur le noyau par toute la matière contenue dans une cavité plate imaginaire, perpendiculaire à l'axe c et où le champ macroscopique : Fig. 2. — Spectres de 13C de la sphère de HOPG. L'origine est la position de la raie de C6H6 (TMS = -130 ppm). Période de répétition : 100 s. Nombre de passages : 1 : 28; ||: 1 700. Fig. 2. 13C NMR Spectra in the HOPG
1 700. Fig. 2. 13C NMR Spectra in the HOPG sphere. O ppm is the C6H6 position (TMS = 130 ppm). Pulse repetition rate: 100 sec. Number of scans: 1 : 28; ||: 1,700. C. R. Acad. Sci. Paris, t. 307, Série II, p. 1693-1697, 1988 1697 est la somme du champ démagnétisant et du champ de Lorentz dus à la matière extérieure à la cavité; Hmacro est le terme le plus important dans le graphite lorsque H0||c, contrairement aux composés d'intercalation dans lesquels la susceptibilité Xn est beaucoup plus petite. Le champ Hj comprend les contributions localisées des atomes
petite. Le champ Hj comprend les contributions localisées des atomes de la cavité ainsi que la contribution des courants délocalisés responsables de la forte susceptibilité Xn Compte tenu du caractère bidimensionnel du système et conformément à une analyse classique [10], nous admettrons que Hmacro représente correctement la valeur moyenne du champ macroscopique (modulé d'un site A à un site B) produit à l'intérieur d'une cavité contenant un unique plan de graphène : Hj est alors, dans cette approximation, le champ produit sur le noyau considéré par les électrons de ce plan. En corrigeant (S|| - 8X) de la petite anisotropie
plan. En corrigeant (S|| - 8X) de la petite anisotropie d'origine dipolaire AK que nous évaluons à + (7-14) ppm, et en posant 5j = Hj/H0, l'anisotropie résiduelle des déplacements (qui ne peut être que d'origine orbitale) d'un plan de graphène est ainsi obtenue à partir de (1) et (2) : (avec Xn = -47,8 ppm, x ± 0,8 ppm et un facteur de Lorentz nul pour H0 || c). 3.3. Ces valeurs se comparent à l'anisotropie des déplacements du benzène gelé : - 180 ppm [2]. Dans ce dernier cas, l'anisotropie résulte principalement des transitions a → 7t* et
cas, l'anisotropie résulte principalement des transitions a → 7t* et 7t → a* qui produisent un important déplacement paramagnétique orbital lorsque H0 || c (terme de Ramsey) [11]. L'extrapolation à densité d'états nulle de l'anisotropie des déplacements dans les composés d'intercalation graphite-alcalin conduit à une valeur du même ordre de grandeur : -190 ppm ([3], [4]). La présente expérience montre que ce terme, de caractère local, est sensiblement identique dans le benzène, le graphite et ses composés d'intercalation et dominerait donc l'interaction orbitale dans un plan de graphène. Cette conclusion est en accord avec des calculs récents ([5], [7], [8])
est en accord avec des calculs récents ([5], [7], [8]) et illustre l'étroite parenté du graphite et des molécules aromatiques. Note reçue le 14 mars 1988, acceptée le 10 octobre 1988. RÉFÉRENCES BIBLIOGRAPHIQUES [1] J. CONARD, H. ESTRADE, P. LAUGINIE, H. FUZELLIER, R. FURDIN et R. VASSE, Physica, 99 B, 1980, p. 521. [2] J. CONARD, P. LAUGINIE, H. ESTRADE-SZWARCKOPF, G. HERMANN, D. GUERARD et P. LAGRANGE, Physica, 105 B, 1981, p. 285. [3] K. KUME, K. NOMURA et Y. HIROYAMA, Synth. Metals, 12, 1985, p. 307. [4] K. KUME, in: Colloque franco-japonais sur les Composés d'Insertion du Graphite, 1985,
in: Colloque franco-japonais sur les Composés d'Insertion du Graphite, 1985, Annales de Physique, 11, Coll. n° 2, Suppl. au n° 2, 1986, p. 245. [5] T. TSANG et H. A. REISING, Solid State Comm., 53, 1985, p. 39. [6] P. LAUGINIE, H. ESTRADE-SZWARCKOPF, B. ROUSSEAU et J. CONARD, Colloque Annuel de la RCP « Composés lamellaires », Le Croisic (France), 1984, non publié. [7] Y. HIROYAMA et K. KUME, Solid State Comm., 65, 1988, p. 617. [8] R. SAITO, K. KOBAYASHI, M. TSUKUDA et H. KAMIMURA, Synth. Metals, 23, 1988, p. 265. [9] J. CONARD, ibid. [4], p. 235. [10]
p. 265. [9] J. CONARD, ibid. [4], p. 235. [10] H. MUELLER, Phys. Rev., 15, 1936, p. 547. [11] L. SALEM, The Molecular Theory of Conjugated Systems, Benjamin, 1966, p. 222. Centre de recherche sur les Solides à Organisation cristalline imparfaite, 1 bis, rue de la Ferté-Bois, 45071 Orléans Cedex 2. L'halogènone choisi a été un iodure d'alkyle car les bromures et les chlorures d'alkyle sont considérés comme non réactifs avec le zinc. La formation d'un organozincique a, dans ces cas, été mise en évidence par sa réactivité vis-à-vis d'un chlorure d'acide. L'addition d'un iodure d'alkyle RI au solvant contenant