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Journal of Materials Science

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01.06.2011 | IIB 2010

A review of some elements in the history of grain boundaries, centered on Georges Friedel, the coincident ‘site’ lattice and the twin index

verfasst von: O. B. M. Hardouin Duparc

Erschienen in: Journal of Materials Science | Ausgabe 12/2011

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Abstract

I trace the origin of the inverse density of coincident lattice sites to Georges Friedel in 1904 (Études sur les groupements cristallins). Georges Friedel (1865–1933), son of the Chemist and Mineralogist Charles Friedel, called this parameter the twin (macle) index and defined it as the ratio of the total number of nodes of the primitive lattice to the number of coincident nodes restored by the twin operation. Friedel’s 1904 ‘multiple lattice’ is our Coincident Site Lattice. Georges Friedel introduced the Σ symbol in 1920 (Contribution à l’étude géométrique des macles) as the ratio of the volume of a (not necessarily primitive) multiple cell to the volume of the primitive cell. G. Friedel provides his reader with several formulae which, in the cubic case, give Σ = h ² + k ² + l ² (h, k and l being the indices of the twin plane) and a twin index I equal to Σ if Σ is odd, equal to Σ/2 if Σ is even. All these definitions and formulae are included in the 1926 version of his celebrated textbook ‘Leçons de Cristallographie’. Georges Friedel was also concerned with the ‘material lattice’ (the crystal structure) behind the mathematical lattice, but besides his contributions to the study of liquid crystals, Georges Friedel was mainly interested in Mineralogy and not in Metallurgy. This may explain why Walter Rosenhain apparently never knew of Friedel’s work and why Kronberg and Wilson had to re-discover the importance of the density of coincidence sites, at the atomistic level, in 1949 in copper. Georges Friedel’s grandson, Jacques Friedel, made the first numerical estimate of interface energies using interatomic potentials that same year but only published these results in 1953. Knowledge of these past events may help us to better understand the present theories and, hopefully, to develop our future understanding more efficiently.

Vorheriger Artikel Grain boundary engineering: historical perspective and future prospects

Nächster Artikel Identification of singular interfaces with Δgs and its basis of the O-lattice

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In French, Georges is spelt with a silent final s.

A process called ‘hydrothermal synthesis’. Robert Bunsen used glass vessels in 1839.

In 1893, in parallel with Henri Moissan, he even thought he might have succeeded in synthesizing diamond.

Charles Friedel had studied at the University and was not a ‘Polytechnicien’. Conversely, the French University always refused to grant Georges Friedel a salary when he taught at Strasbourg University after 1919. This dual French school system still exists today.

Having spent his youth in an apartment in the building of the School of Mines, where his parents lived since his father also was the curator of the mineralogical collection, Georges had first expressed the wish to specialize in a quite different field, namely Naval Architecture, after the École Polytechnique. Yet, being rated first, he was not free to choose.

G. Friedel strongly objected to the inappropriate term ‘liquid crystal’ (Otto Lehmann’s Fliessende Krystalle) but this appellation remained.

First noted by Nicolas Steno, or Nils Stensen [‘Son of Stone’], (1638–1686), an outstanding anatomist who also layed down the fundamental principles of stratigraphy, and thus of geology. Steno later ruined his health in Catholic missionary work. He was beatified in 1988. His ‘preamble about solids naturally contained within solids’ (De solido intra solidum naturaliter contento dissertationis prodromus, that is, an attempt to explain the formation of fossils), published in Florence in 1669, remained largely unknown. To give full support to the law of constancy of inter-facial angles for minerals of the same species, the technical invention of the contact goniometer by Arnould Carangeot, Romé de l’Isle’s assistant, was essential. About goniometers, see [22].

Note the variability of the orthography: crystaux, cristaux, cristallen, Krystalle, Kristalle. The Greek root ‘krustallos’ meant ‘solidified by cold’ (kruos: see cryogenics). In ancient times quartz was believed to be a permanently solidified form of ice. Robert Boyle was the first to use the word crystal in a general sense, not restricting it to rock crystal, in The Sceptical Chymist (1661). The German word Quarz is presumed to be of Slavic origin, although its exact meaning is not known.

With the common etymology of zwei and two, akin to duo, double, duplex, dyad, as well as deux in French. Macle was also used in English, for instance by Lord Kelvin of Largs, in a Robert Boyle Lecture delivered at the Oxford University Junior Scientific Club, on the evening of May 16, 1893 (and reproduced as Appendix H in the Baltimore Lectures): ‘Coming back to quartz, we can now understand perfectly the two kinds of macling which are well known to mineralogists …’.

The French also use the name ‘joint de grains’ = grain boundary (Korngrenze). For the etymology of macle, Webster’s dictionary gives the heraldry term mascle which corresponds to an empty lozenge, from the middle-old Dutch mask and maesche, which also gave mesh (= Maesche = maille). Hence, sometimes, a circumflex on the a: mâcle (for instance Pierre Curie in 1900, [17]). Recent comprehensive French dictionaries like Le Grand Robert (1985) and the Trésor de la langue française (CNRS & INLF Nancy, 1985) give the same etymology. The Trésor writes both macle and mâcle (mâcler).

As a Teacher of Grammar at the Cardinal Lemoine College in Paris, René-Just Haüy got acquainted with Charles Lhomond, a famous grammarian who was very keen in botany. In order to please his Friend, René-Just learned the names of hundreds of plants and this certainly later helped him name crystals and groupings of crystals. René-Just Haüy had no inherent taste for Botany, but his walks with Charles Lhomond in the Jardin du Roy, adjoining their College led him to attend the lectures of Louis-Jean-Marie Daubenton who also taught Mineralogy.

But from a purely aesthetical view point, I am sure that all would have been delighted at the sight of a drawing by Albrecht Dürer in his manual on measurement of lines, areas and solids by compass and ruler [27]. This drawing looks like a (double) five-fold twin, almost quasiperiodic (a size-limited approximant in fact), see [28] and [29]. Figure 2.3, by Eric Lord, Alan Mackay and S. Ranganathan, has a perfect pentagonal shape, much akin to the characteristic hexagonal shape of multiple twinning in aragonite. It is a pity that multiple-twinning led Linus Pauling (1901–1994) to reject the discovery of quasicrystals in 1985 [30]. Yet that challenge was not illogical and certainly stimulated refinement of the arguments in favour (the pigeons pro eventually won versus the contra cat, see [31]). Linus Pauling tried to contact with Danny Shechtman but unfortunately no satisfactory agreement was reached before Pauling’s death (D. Shechtman, personal communication, July 2010).

Even if his father was a proponent of the existence of atoms, as was his mentor Adolphe Wurtz (1817–1884), it was clear that little was known about their actual organization within a crystal until the advent of X-ray diffraction in 1912. It is worth noting that ‘Friedel’s law’ in X-ray crystallography was established by Georges Friedel as early as 1913. [32]. This law states that the intensities of the Laue-Bragg diffractions I(Q) and I(-Q) are equal under normal conditions. That is, the information obtained by diffraction is centro-symmetric and cannot establish whether the real-space atomic distribution is centro-symmetric (with a point of inversion) or not. This also means that diffraction spots appear in pairs which have recently been called ‘Friedel pairs’: Friedel pairs are key to both the efficiency and accuracy of X-ray diffraction contrast tomography, which permits non-destructive mapping of grain shape and crystal orientation in polycrystals. [33].

With the rejection of many chance crystal groupings that may look like twins at first sight but are actually aberrant twins (‘macles aberrantes’). For instance, ‘specimens noted I and III [in quartz] by Zyndel would be [according to Zyndel] twinned according to the La Gardette law (Japanese law). It suffices here to look at the sample with the naked eye, without any measuring and just having its faces glance, to realize that such is not the case’. [34].

Many German scientists also contributed to the descriptions and analyses of twins: Christian Samuel Weiss, Friedrich Mohs, Carl Friedrich Naumann, Gustav Rose, Friedrich Eduard Reusch, Heinrich Adolf Baumhauer, Otto Mügge, Gustav Tschermack, Victor Moritz Goldschmidt, Jakob Beckenkamp, not to name all. Georges Friedel opposed many of them, but not because France had lost in Sedan in 1870: Georges Friedel also opposed his compatriot Frédéric Wallerant [35], and criticized Ernest Mallard when their views differed.

Mero means part in Greek, as opposed to holo meaning whole. Hedra means seat, base or face. Hemi means half, tetartos means four and ogdoas eight. When hedra comes with a prefix it usually looses the aspired h (a special diacritic sign on the Ε in Greek which is then pronounced by rough breathing). This explains the French and German spellings of for instance: polyèdre, holoèdre, polyeder and holoeder. Modern English spelling keeps the h and writes polyhedron and holohedry, etc. French writes mériédrie, although méroédrie would be slightly more correct.

Only the orientation of the atomic unit (basis) within the primitive cells changes. These twins have a twin index (see below) equal to 1.

Auguste Bravais first wrote with his elder brother, Louis: an Essai géométrique sur la symétrie des feuilles curvisériées et rectisériées. This was transmitted to the French Academy of Sciences before 1837, see [36].

‘But it is clear that this is a purely fictional move’ (‘mais il est clair que c’est là un mouvement purement fictif, et que la coordination moléculaire se fait symétriquement par rapport au plan d’hémitropie’), said Auguste Bravais in 1850 [37]. For instance, the Σ = 3 {111} 〈110〉 θ ~ 70.53° twins in cubic systems can also be described as hemitropic Σ = 3 {111} 〈111〉 (θ = 180°) (as well as Σ = 3 {111} 〈111〉 θ = 60° twist grain boundaries since 〈111〉 are threefold axes in cubic systems).

In 1885 [38], not in 1876 [25].

As is the case for the twins by merohedry described previously.

The primitive lattice point group is not necessarily a subgroup of the coincidence lattice point group, because it may possess symmetry elements that are not shared by the coincident lattice (see [40] and [41]).

These twins are frequent in minerals which have a complicated atomic group associated with each primitive lattice point or node.

With a natural extension to cases of twins formed by pseudo reticular merohedry. See Table 2 for a synthetic presentation of the four classes developed by Friedel.

Friedel noted that the (310) case had never been observed.

A School to whose development he significantly contributed. He became Head of the School from 1907 to 1919, except for the war years.

The Friedels were Alsatian from Strasbourg. As a first name, Friedel is a variant for Gottfried. The German Friede means peace.

See Appendix C for some further considerations on simple analytical properties of the twin index in cubic crystals.

It is remarkable that G. Friedel immediately understood the principles of X-ray diffraction discovered in 1912, and explicited the inversion symmetry limitation (under normal conditions) known as Friedel’s law ([32], see also footnote 13). Donnay and Harker were able to expand Friedel’s 1911 preview for surfaces in 1937 [46]. Friedel’s formalism is limited to a consideration of Bravais lattices and does not incorporate the Schönflies-Fedorov space groups. Denis Gratias, Richard Portier, Robert Pond and others further extended Friedel’s formalism for twins in the 1980’s, but this goes beyond the scope of this article. Also see Appendix D.

In contrast, the French(‘La Gardette’)-Japanese twin in quartz is much rarer and complicated, see [48, 51‐54]. There are many other twin laws for quartz. One of these is also known as the Friedel-law, or Friedel-twin, having been found by Charles Friedel in artificial quartz in 1888 [55].

Gregori Aminoff and (his wife) Birgit Broomé proposed several rules in 1935 about the atomic structure of twins in minerals. These rules have been reported by Robert Cahn [39]: 1. When two individuals form a contact twin either one or two layers of the structure at the interface are common to both individuals. 2. The atomic coordination in the transition layer is either (almost) identical with that in the crystal structure or closely related to it. In the latter case, the transition structure is that of a possible polymorphic modification of the structure, or else that of a modification which would be possible for that substance. British physicist by heart, R.W. Cahn was born in Germany and could speak French, see [57]. His 1954 review article is a wealth of informations about twins. .

The order n of a Σ = 3ⁿ twin (or any Σ = Σ _o ⁿ twin) should not be confused with the twin index Σ itself. Such a confusion can arise because n has also been used as a symbol for the twin index (by G. Friedel himself, in his textbooks. In 1926 one finds n and I), and Paul Niggli (see Appendix B) translated Friedel’s French word indice by Ordnung in his books in German ([63, 64]).

These twins thus merit study at the atomistic level, and have been investigated by joint numerical and observational studies, see [65, 66]. These twins are called grain boundaries in these studies. See, however, Appendix E.

For instance William Lawrence Bragg, Bragg junior, could derive the fcc structure of copper as soon as 1914 thanks to natural crystal specimens from the Mineral Laboratory at Cambridge [67]. Other metal samples, usually consisting of tiny crystalline grains, had to await the development of the powder diffraction technique in 1917 (Peter Debye and Paul Scherrer in Göttingen, Germany, Albert Hull in Schenectady, in the US).

Compare the following with Ronald King and Bruce Chalmers [68], Chaps 1 and 2 of Donald McLean [1], Ernest Hondros’ ‘enquiry’ in 1995 [69], and David Brandon’s recent perspective [70].

Typical is the first sentence of HC Sorby in his article ‘On the microscopical structure of iron and steel’, published in 1887: ‘It is now more than 20 years since I first commenced to carefully study the microscopical structure of iron and steel, in order, if possible, to throw light on the origin of meteoric iron; but soon found that the results were of even more value in connection with practical metallurgy’. [72]. Most ceramic materials are also manufactured, but rarely from the melt and, at first glance, look more like hard minerals than like soft metals.

Rosenhain had been thinking about the problem for several years, presumably since 1904, as would appear from the discussion of a paper by GD Bengough [82] where Guy Bengough (1876–1945) wrote that ‘the first action of a dilute reagent is to eat into the crystalline boundaries’ so that ‘the deduction may reasonably be drawn that the individual crystals in a pure metal are normally bound to one another by some substance stronger than the crystals themselves, but more easily attacked by etching agents. This substance must surely be no other than Beilby's amorphous material, arranged in a thin, more or less continuous layer round the crystals’. Sir George Beilby (1850–1924) never reacted positively to this hypothesis concerning the structure of grain boundaries.

Although he did teach ferrous metallurgy in the 1890’s at the École des Mines in Saint Étienne.

Speaking of Anderson and Norton, Foley wrote: ‘The authors have apparently driven another spike in the coffin of the general amorphous-metal hypothesis which has been reared a weakling from its inception’.

That is, the (ductile) intergranular fracture of metallic polycrystals at high temperatures versus the (brittle) transgranular fracture at low temperatures (when the amorphous interface can be assumed to be as hard as a glass below its transition temperature).

‘such positions as will balance the atomic forces’. This over estimation of the amplitude of atomic relaxations probably allowed them to consider that, under stress the interface would become amorphous so that ‘the material will behave in the manner described by Rosenhain in connection with the amorphous cement theory’. Yet, even in the unstressed condition, Rosenhain wrote in the discussion that he could not accept their illustration: ‘I think that it implies an arrangement of atoms in a condition which I think is not one of possible stable equilibrium. It implies atoms being brought in some places too close together and in others too far apart to fulfil what we believe is known of the conditions of atomic linkage that exist in solid metals. Such an arrangement of atoms is certainly improbable and would require proof before one could accept it as a fact’. Recent observations and simulations of asymmetrical grain boundaries in copper show that atomic disorder exists only locally [65, 66]. A true \( (100)_{1} //(\bar{4}30)_{2} \) tilt GB has recently been grown, observed and simulated, in a ceramic material: SrTiO₃ [90].

Thum saw the development of slip planes but he unfortunately failed to recognize dislocations (which were probably present) because he had not been expecting to see them. Bubble soap models were rediscovered by Sir W. Lawrence Bragg and John F. Nye in 1947. With respect to Rosenhain’s model, dislocations did not help to simplify the considerations of deformation mechanisms in polycrystalline metals, but they are necessary to explain in details what is observed because they correspond to reality. At about the same time, the early thirties, the neutrino was postulated, not to simplify the existing theories, but to explain the riddle of the observed continuous energy spectrum of nuclear beta electrons.

G. Friedel proposed the term nematics, from the Greek nema meaning thread, ‘because of the linear discontinuities, which are twisted like threads’, (‘à cause des discontinuités, contournées comme des fils’ [21]). These topological defects correspond to disclinations. For discontinuities in smectics, G. Friedel and his son Edmond (see Table 1) noted in 1931 that they must have ‘the form of groups of focal conics’ (‘les discontinuités n’y peuvent apparaître que sous la forme d’un groupe de coniques focales’ [91], See also G. Friedel and François Grandjean in 1910/1911 [92, 93]). François Grandjean (1882-1975) later became a specialist of acarians.

This idea was generalized by Nevill Mott in 1948: ‘If two crystal planes are in contact, but cannot fit owing to different indices of orientation, one may suppose that the surface of contact is divided into islands where the fit is reasonably good, separated by lines near which fit is bad’. [94].

The same reasoning applies for twin orientations near the special {111} twin in elemental fcc crystals, and as well as for the contribution of widely spaced secondary dislocations, so, as noted by Thornton Read and William Shockley, misorientations near energetically favoured twins may have the characteristic |δθ| ln|δθ| additional cusp contribution [96]. This does not work near the structurally favoured {122} twin, the energy of which does not correspond to a cusp.

Jacques Friedel went to Bristol (1949–1952) to work with Nevill Mott and learn more about the electronic interactions in metals. Some of Jacques Friedel’s later contributions in that field eventually led, with François Ducastelle, to the development of the Finnis–Sinclair potentials for transition and noble metals (see [99]). His stay in Bristol also led Jacques Friedel meeting his future wife, a younger sister of Nevill Mott’s wife. He also met there with Charles Frank, who thus learnt of Georges Friedel’s Leçons.

Friedel wrote that what mostly prompted this extention was a ‘beautiful work’ by M. Schaskolsky and A. Schubnikow using in 1933 crystals of alum in an experiment very much akin to the later MgO smoke experiments. This Schubnikow is no one else that Alekseï Vasil’evich Shubnikov (1887–1970) who is famous for the Shubnikov groups, or antisymmetry groups, magnetic groups, coloured groups, which have been used in many fields, including the study of Grain Boundaries (for instance by Yves Le Corre and Hubert Curien in 1958, and by Denis Gratias and Richard Portier, and Demosthenes Vlachavas and Robert Pond in the 1970s and 1980s). Shubnikov had also been a pioneer in the formation and growth of crystals, in Leningrad (St Petersburg) and in Moscow.

Denis Gratias and Richard Portier already complained in 1982 that ‘the terminology to day in common use in grain boundary community is often unfortunate and confuses, for example, lattice nodes and crystal sites’. [117].

E.g. Charles Kittel, Introduction to Solid State Physics, pp. 4–5, Neil Ashcroft and David Mermin, Solid State Physics (1976) p. 75. It was introduced, in German, by Max Born in 1922 [118]. At the Göttingen University Born served as an assistant to David Hilbert, his mentor, who was famous for his ‘basis theorem’ so that Born was certainly aware of the first usage of the term basis.

Although, as we have seen, it is a sublattice from a mathematical point of view.

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Titel: A review of some elements in the history of grain boundaries, centered on Georges Friedel, the coincident ‘site’ lattice and the twin index
verfasst von: O. B. M. Hardouin Duparc
Publikationsdatum: 01.06.2011
Verlag: Springer US
Erschienen in: Journal of Materials Science / Ausgabe 12/2011
Print ISSN: 0022-2461
Elektronische ISSN: 1573-4803
DOI: https://doi.org/10.1007/s10853-011-5367-1

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