STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VARIATIONS VERSUS THE TEMPERATURE J. Sadoc, R. Mosseri To cite this version: J. Sadoc, R. Mosseri. STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VARIATIONS VERSUS THE TEMPERATURE. Journal de Physique Colloques, 1982, 43 (C9), pp.c9-97-c9-100. <10.1051/jphyscol:1982918>. <jpa-00222447> HAL Id: jpa-00222447 https://hal.archives-ouvertes.fr/jpa-00222447 Submitted on 1 Jan 1982 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
JOURNAL DE PHYSIQUE Colloque C9, supplément au n 12, Tome 43, décembre 1982 page C9-97 STRUCTURES OF AMORPHOUS MATERIALS AND SPECIFIC VOLUME VERSUS THE TEMPERATURE VARIATIONS +. * J.F. Sadoc and R. Mosseri +Physique des Solides, V.P.S., Bâtiment 510, 91405, Orsay, France *Physique des Solides, C.N.R.S., 1 Place A. Briand, 92190 Meudon-Bellevue, France Résumé. - Une description systématique de la structure amorphe est présentée et utilisée pour expliquer les variations de volume spécifique dans les composés amorphes lors des variations de température. Abstract. - A systematic structural description of amorphous materials is presented and used, to explain the variation of the specific volume of amorphous (or glassy) compounds versus temperature. Introduction. - Thermal variation of the specific volume of elements or simple compounds generally exhibit a small increase with T in the solid state, a discontinuity at the melting point and an important increase for the liquid specific volume at higher temperatures. Nevertheless if the compound is in an amorphous state the specific volume variation curve is continuous and can be divided in two regions. At low temperature the specific volume is a few percent higher than in the crystalline state. The two curves showing the variation of the crystal and the glass specific volume are approximately parallel. In the second region, at higher temperatures, the.glass specific volume curve is in continuity with the undercooled liquid one. The narrow transition zone between the two parts is generally explained in terms of a glass transition. In this paper we interpret this behaviour with pure topological (structural) arguments. Amorphous structures described as regular structures in curved spaces In various papers (1 )(2 )(3 ) we have presented a new description for amorphous structures: amorphous structures are supposed to appear if a local order induced by local chemical binding is uncompatible with periodicity requirement (for example in the case of a 5-fold local symetry). But a motive of a given local order can be arranged regularly in a non-euclidean space having a constant curvature (spherical or hyperbolic space). A map of the curved space onto the euclidean space introduces topological defects in the regular structure and leads to a nonperiodic structure. These defects can be of different types, for example internal surfaces or disclination lines associated with elastic distortions. These defects can be also described in terms of added volume, as they change the density while they allow a decrease of the space curvature. The density of close packed structure We limit our presentation to close packing structure but we think that it can be extented to other structure. It is well know that the crystalline close packing structure (f.c.c.) have a packing fraction p = 0.74. Nevertheless, since there is octahedral and tetrahedral holes in this structure we can imagine a more dense structure in which there is only tetrahedral holes. As the tiling of euclidean Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982918
C9-98 JOURNAL DE PHYSIQUE space with regular tetrahedra is impossible there is not any periodic structure with this property. But such a regular structure can be described in a spherical curved space (3 ). In this case all holes are tetrahedra and the local coordination polyhedronis a regular icosahedron. This structure is an appropriate starting point tor the description of mono-atomic amorphous metallic structures. The packing fraction of this structure is equal to 0.774 which is greater than the f.c.c. value (2). But this structure is defined in a curved space. Decrease of the curvature requires defects which lower the density. In dense random packing of hard spheres it is found experimentally ( 5 ) and theoretically ( 2 ) that the packing fraction is 0.63. If atoms interact with a more realistic potential than the hard sphere potential, a1 lowing extension and compression of the interatomic distance, then p = 0.70 is a more accurate value. For instance this value is observed after relaxation of the hard sphere packing with a Lennard-Jones potential ( 6 ). Specific volume variation vs temperature in terms of the curved space description There are numerous specific volume data concerning metals. Here we use the copper as an example, but the copper results can be extented to other metals. The specific volume variation given by different authors (7 )(8 ) is presented on fig.la for both the liquid and the f.c.c. copper. The data are normalized to the atomic volume at T = OK. With this unit V (0) = 1/0.74 for the crystalline specific volume at 0 K taking into account the f.c.c. packing fraction p = 0.74. The dashed curve shows the extrapolation of the specific volume of the liquid down to low temperature. As yet observed by Kauzmann in its original paper the liquid appears to strive for a smaller specific volume than the crystal at temperatures well above 0 K. The extrapolated specific volume for the liquid copper tends at 0 K to the value v~(0) = 1/0.769 (with the same unit as Vc(0)). It is striking that this value is very close to the specific volume of the structure defined as the ideal dense structure in curved space v1 = 1/0.774 (taking into account the above mention packing fraction p = 0.774). This behaviour observed for copper is also present for argon or some other simple metals. - The liquid specific volume The behaviour of the liquid specific volume, and of its extrapolation to low temperature can be explained using the hypothetical structure above defined in a curved space. In this structure, we suppose an increase of the number of defects when the temperature increases. This defect density variation is responsible for the regular variation of the specific volume from 0 K to the liquid state temperature. But in this description the structure remains in a curved space. If we suppose that the defects can change the curvature (which is the case for disclinations or internal surfaces), an increase of the temperature is associated with a decrease of the curvature. For a given temperature To the number of defects is sufficient to achieve a complete decrease of the curvatureandthe actual structure belongs to the euclidean space. For temperature over To the increase of the number of defects does not change the curvature but correspond to a change of the volume allowing diffusion process and leading toanusual liquid behaviour. Under To the change of the volume is associated with a decrease of the space curvature,over To the "added volume" can be called "free volume". Notice that is it only over To that the described structure can be physically observed. - The glass (or amorphous) specific volume Looking at an hypothetical amorphous copper we can easily predict its specific volume variations using structural arguments. Suppose the structure to be a dense random packing of sphere. As we said above this structure can be described by mapping a curved structure onto the euclidean space. The packing fraction (p = 0.70 with soft interaction) of this structure leads to a reasonable value of the specific VOlume of the amorphous state at T = 0 K to v~ (0) = 1/0.70. As pure amorphous copper is not experimentally observed there is no specific volume measurements but this value is of the order of what is observed in most amorphous metallic alloys.
The thermal variation of the volume in the solid state does not depend on topological change,but only on the asymmetry of the pair interaction between atoms.it is the reason why we suppose the same variation for amorphous and the crystalline specific volume. From these hypotheses it results that the amorphous specific volume starts from VG (0) = 1/0.70 at T = 0 K and slowly increases like the crystalline specific volume. Comparison between the liquid and the glass specific volume The experimental liquid copper specific volume versus temperature extrapolated for the undercooled liquid, is ploted on the fig.lb. The estimated variation for the hypothetical amorphous copper is also presented. These two curves showing the liquid and the amorphous (or glassy) behaviour intercept for the temperature To. Indeed at this temperature the amorphous solid state and the liquid state are described by the same model: the curved space model with exactly the optimum number of defects allowing a complete decrease of the curvature to zero. Conclusions and discussions In numerous papers the density variations of the liquid and amorphous materials are explained by an extrapolation of the liquid density to low temperature, with a change in the slope of the curve when the extrapolated density reaches the crytalline one (9)(10). This ideal case occuring only at an extremely low cooling state. Consequently in all these models an ideal amorphous structure with a density equal to the crystalline structure is (more or less implicitly) supposed. In other terms it is possible to go continuous~y from a real amorphous structure to the crystalline one. In the present model we also consider ideal amorphous structures. But two ideal structures are defined. The first one is a perfect regular structure (with the amorphous local order) defined in a space of a given curvature. The second one, is the ideal structure obtained by mapping the last one onto the euclidean space with an optimum (minimum) density of defects. These structures are completely different from the crystalline one, as they have not the same local order. The structure defined in euclidean space also differ from a crystal because it contains necessarily defects. It can be considered as the basic structure to which all real structures can be compared. The existence of such structure is related to a very low cooling rate. However such a cooling rate would not prevent crystallization. Consequently real amorphous structures contain a number of quenched defects greater than the optimum value resulting from the finite cooling rate. Extrapolation of the liquid structure to very low temperature leads to the ideal structure defined in curved space. A similar approach has been already used to explain qualitatively the density variation in a Si-H compound versus the H content (11). The intercept of the two curves showing glass and liquid specific volume variations allow the definition of an ideal glass transition temperature To independantly of the crystal line solid state. In a recent paper Jackle (12)gives two formulations of the glass: glass as a quenched liquid or glass as a disordered solid. The present description associates these two formulations since the two models describing the glass structure and the liquid structure are identical at the temperature To. Notice that this approach is, in a topological sense, close to the Edwards(l3,14)approach using dislocation as a parameter which is frozen in the amorphous state. Here the parameter is related to an other kind of defect (for example it can be disclination with a parameter q defined as the length of disclinaiion in a unit volume). Nevertheless the Edwards approach using a dislocation model for the liquid and the amorphous state implicitly suppose the continuity between the crystalline and the amorphous state in contrast with the present model. Following our hypothesis it is easy to understand why the "q" parameter is frozen: it is due to absolute necessity for the structure to belong to the euclidean space which requires a minimum number of defects. This description of the amor~hous and the liquid state gives an answer to the Kauzmann paradox, "which would imply the existence of some End of state of h-igh
C9-100 JOURNAL DE PHYSIQUE order for the liquid at low temperature which differs from the normal crystazzine state" (W.K.). We have in this paper used this approach to explain the specific volume behaviour of a simple metal in liquid and amorphous state. But it is possible to extends it to other properties (specific heat, compressibi 1 i ty, etc...) or to other elements or compounds. Fig.1-a) Variation of the copper specific volume from experimental results for the liquid (R) and the crystalline (c) state. b) txtrapolation to low temperature of the liquid variation. This variation corresponds to an under cooled liquid (u). kstimation of the specific volume variation for an ideal glassy copper (g). The unit for volume is the volume of a spherical atom of copper. p is the packing fraction. BIBLIOGRAPHY (l)m.kleman, J.F.Sadoc, J. de Phys. Let. 40 (1979) L 569. (2)J.F.Sadoc, J. Non Cryst. Sol. 44 (1~81F1. (3)J.F.Sadoc, R.Mosseri, Phil.Mag. B, 45 (1582), 467. (4)H.S.M. Loxeter, Regular Polytopes, Dover publications New York (1973). (5)J.L.Finney, Proc. R. Soc. A 319 (1970) 4/9. (6)F. Lan~on, L-Billard, J.Laugier, A.Chamberod, J. Phys. F.: Pet.Phys. 12 (1982)259 (7)K.Bornemann and Sauerwald, Z. Metall. 14 (1922) 145. (8)A. tl Mehairy and R.G. Ward, Trans. Met. Soc. AIFIE Z27 (1963) 1228. (9)W. Kauzmann, Chem. Rev. 43 (1348) 219. (lo)c.t. Moynihan, A.J. Eastel and M.A. De Bolt, J. of the Am. Cer. Soc. 59 (1976)12 (ll)j.f.sadoc, R. Mosseri, Jour. de Phys. C4, sup.no 10, 42 (1981) C4-189. (12)J. JSckle, Phil. Mag. B, 44 (1981) 533. (L3)S.F. Edwards, Polymer 17 (1976) 933. (14)N. Rivier, Jour. de Phys. C6, sup.nv 8, 39 (1978) C6-984.