Mechanical and structural properties of metallic glasses HAMIDA. RAFIZADEH The Dayton Power and Light Company, P.O. Box 1247, Dayton, OH 45401, U.S.A Received August 26, 1988

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

A simple linear-chain model is utilized in calculation of the mechanical and structural properties of metallic glasses. The model calculations of composition-dependent elastic constants show good agreement with the experimental data available for binary metallic glasses. The model is capable of distinguishing different structural characteristics of glassy systems and provides excellent agreement with the radial distribution functions of amorphous cobalt, metal-metalloid, and binary transition metal glasses. The model is not limited to binary systems and can be readily applied to ternary and higher order metallic glasses. On utilise un modele simple B chaine linCaire pour le calcul des propriCtCs mCcaniques et structurelles des verres mCtalliques. Les calculs faits sur ce modkle pour les constantes Clastiques dependantes de la composition donnent un bon accord avec les donnees expCrimentales disponibles pour les verres mCtalliques binaires. Le modkle est capable de distinguer diffkrentes caractCristiques structurelles des systkmes vitreux, et il donne des accords excellents avec les fonctions de distribution radiale des verres au cobalt amorphe, des verres metal-metalloide et des verres binaires B mitaux de transition. Le modkle n'est pas limit& aux systkmes binaires et peut facilement &tre applique B des verres mCtalliques ternaires ou d'order supCrieur. [Traduit par la revue] Can. I. Phys. 68, 023 (1990)

1. Introduction The structural characterization of metallic glasses encompasses a relatively wide range of crystalline, quasi crystalline, and dense random packing models (1). Although there is some degree of consensus as to the existence of short-range order in the metallic glasses, there has been considerable variation as to the selection of building blocks or structural units for the glassy models (2). Currently, most of the structural characterizations of metallic glasses have been accomplished through models that utilize some form of dense random packing of atomic positions (3,4). Unlike the structural properties, the modeling of the mechanical properties of metallic glasses has been relatively limited. One approach utilized by Weaire et al. (5) employs the radial distribution function in the derivation of expressions for the interaction energy and elastic moduli, achieving general qualitative agreement with the experimental observations. Another theoretical study (6) utilizes a tight binding approximation for the attractive part and a Born-Mayer type potential for the repulsive part of the interaction energy in order to calculate elastic constants of amorphous transition metals. This approach has produced qualitative features of glassy metal elastic constants. In this paper, we intend to focus on the application of simple linear-chain models for the calculation of elastic constants and structural properties of metallic glasses. The application of a linear-chain model to an amorphous solid is not new and has been extensively utilized in thgstudy of the spectral properties of amorphous systems (7). In this study we will demonstrate that a simple linear-chain model of metallic glasses is a modeling app~oximationthat provides both qualitative and quantitative agreements with a variety of experimental observations on elastic constants and radial-distribution functions. In Sect. 2. the linear-chain model of a metallic glass is introduced. In Sect. 3, expressions for composition-dependent Young's moduli are obtained for binary transition metal glasses and the calculations are ~ 0 m ~ a r ewith d the available e x ~ e r i mental data. Section 4 includes a calculation of the structural

properties for amorphous cobalt and a number of metallic glasses, and the comparison with experimental observations. Section 5 gives the discussion and concluding remarks.

2. Theoretical model The linear-chain model, the simplest model of a solid, comprises a linear lattice with an atomic building block or structural unit asosciated with each lattice point. Starting with general building blocks such as the three-dimensional Bernal structures (8) shown in Fig. 1, the linear-chain (LC) model can be represented as

where n is the number of building blocks in the chain and Bi is the Bernal unit. If the structure is binary, then

where the Bernal units Bi and Bj consist of different types of i and j atoms, respectively. The building blocks can also be made from two-dimensional or one-dimensional structures. A two-dimensional Bernal-type structure is equivalent to a generalized Penrose structure (9) as shown in Fig. 2. The Penrose structure is essentially a distortion of the rhombus, which constitutes the planar close packing unit of crystalline metals. The Penrose structure can be used to construct linear-chain models of the binary system in the form of

u

Prinled in Canadailrnprirn4au Canada

where Piis the Penrose structure consisting of ith type atoms. Figure 3 shows the structure of an LC model for a pure glassy

CAN. 1. PHYS. VOL. 6 8 . 1990

BERNAL'S CANONICAL HOLES

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

TETRAHEDRON

HALF-OCTAHEDRON

T R I G O N A L PRISM

TETRAGONAL DODECAHEDRON

FIG. 1. The three-dimensional building blocks that can be utilized in construction of linear-chain models.

metal that utilizes the two-dimensional Penrose structure as its building block. The Penrose structure possesses interesting features as a building block for metallic glasses. The Penrose units, as shown in Fig. 4, can be put together to model metal-metal or metal-metalloid structures. For metal-metalloid structures, it has been shown (10) that the larger metal atoms, capable of admitting a smaller metalloid atom in the holes among themselves, have a ratio of approximately one hole per four large atoms of the glassy material. This characteristic is retained by the two-dimensional Penrose structure. Studies of two-dimensional glasses (1 1) have been used to demonstrate that the Penrose structure can fill the two-dimensional space creating perfect local order without having long-range translational order, which is another characteristic of metallic glasses. The simplest of the LC models utilizes a one-dimensional building block, which consists of a string of atoms on a straight line. For a binary system, the model becomes random

[4]

LC =

[(O.. .O), .. . (0.. .),I

8,

where 0.. .O and 0 . . .O represent monatomic linear segments of types i and j, respectively, repeated n and rn times in the LC model. Notwithstanding the simplicity of [l]- [4], a local crystalline arrangement of building blocks along the chain would limit the LC models to numerical calculations based on computer simulation of chains with finite lengths. Since the models have a locally defined crystallographic orientation, but with only

FIG.2. The two-dimensional building blocks. (a) the rhombus of planar close packing in crystalline metals. (b) the Penrose structure that is constructed from distortion of the crystalline rhombus.

short-range translational order, the main distinguishing feature of these models, when compared with crystalline models, is the broken translational variance. Studies of two-dimensional glasses (1 2) in models consisting of two different sized spheres produce a disordered structure that nevertheless shows a stnking sixfold symmetry in the structre function. This indicates that supercooled systems such as metallic glasses develop extended orientational correlations characteristic of a crystalline solid, even in the presence of a broken translational symmetry. Similar observations have been made on the experimental data of many amorphous metals that exhibit crystalline features (13). To summarize the above, we believe that the LC model given in [I]-[4] has a pseudocrystalline character that can be exploited in the modeling of metallic glasses by replacing the random arrangement of the local crystalline units with a periodic arrangement in the form of [5al [5bI [5cl

7[

periodic

LC =

B, ... B, pi ... P, (0.. . q i . . . (0.. .a),

1

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

RAFIZADEH

25

FIG. 3. A linear-chain model construction using the Penrose building block.

translational symmetry has little effect on the results and that the simple LC models are capable of achieving surprisingly good qualitative and quantitative agreements with the experimental data.

FIG. 4. The Penrose building blocks of binary metallic glasses. ( a ) two-dimensional metal hole, (b) two dimensional metal hole with metalloid atom located in the interior of the hole, and (c) the twodimensional metalloid hole.

This approximation, in effect, creates a simple crystalline model of the metallic glasses when compared with other crystalline and quasi-crystalline models (1,13). The basic advantage of the proposed model over other models of the glassy solids lies in its simplicity and ease of application to calculations of a variety of physical properties. In the following sections we will demonstrate that the absence of broken translational symmetry and its replacement with full

3. Calculation of mechanical properties Our starting point in the analysis of the mechanical properties will utilize the simplest of the linear-chain models, i.e., [5c]. In light of limited experimental data, this will minimize the number of adjustable force-constant parameters in model calculations. In these calculations we will distinguish between two major subdivisions of the metallic glasses (14). In the first grouping, the glassy system is a metal-metal structure that can be characterized as a random spatial repetition of large local units, where each local unit will have short-range orientational and translational symmetry. The second grouping of metallic glasses consists of metal-metalloid compositions. The structure of metal-metalloid glasses will possess the same type of local order for the metal atoms as that of the metal-metal glasses, but with the smaller metalloid atoms being interstitially located in the holes of the metallic superstructure. Using [5c], the model given in Fig. 5 represents the metal -metal LC model. The minority metal is represented by a simple two-atom unit surrounded by the many-atom unit of the majority atom. The number of atoms in the repeating unit of the majority metal will be allowed to vary to simulate different compositions. The LC model corresponding to metal-metalloid glasses is given in Fig. 6. Each atom of metalloid, the minority atom, is isolated from other metalloid atoms by the many-atom units formed by the metal atoms. The LC model of metal-metalloid glasses has been successfully applied to the calculation of the metal-metalloid Young's moduli in a demonstration of the complex force-constant dependence of elastic constants (15). In this paper we will focus on the development of expressions for the elastic constants of metal-metal glasses. The unknowns of the model are the force constants of the interaction between atom k' in unit cell 1' and atom k at unit cell 1 defined as the matrix [+](lk, l'k'). It can be shown that [+](lk, l'k') can also be expressed in the form [+I(/', kk') with atom k at the origin.

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

C A N . J . PHYS. VOL. 68. 1990

FIG. 5. The repeating unit of a linear-chain model for metal-metal glasses utilizing one-dimensional building blocks. The interatomic separations a and b are taken to be the same as the nearest-neighbor distances of the corresponding crystalline constituents. The distance c is taken as the average of a and b.

will pursue the derivation of the expressions for Young's moduli involving the force constants In the following, we will not provide details of the calculations but refer the interested reader to ref. 15. Utilizing the convention of Fig. 5 for identification of atoms, we define the force-constant ratios as

+,,.

1

I

FIG. 6. The repeating unit of a linear-chain model for metal-metalloid glasses utilizing one-dimensional building blocks. The interatomi; separation b is taken to be the same a i the nearest-neighbor distance of the corresponding crystalline constituent. The distance c is taken as ( a + b)/2, where a is the nearest-neighbor distance of the crystalline metalloid element.

[9]

Rl

=

+,,(AB) &;@A) '

R'

=

+,,(BB) +;;(AA)

and using the microscopic theory of elasticity, it can be shown (15) that the expressions for Young's modulus, E, for the metal-metal LC model, is given as r

The elements of the force-constant matrix, 161,consist of nine parameters for each pair of interacting atoms; they are

I

The number of unknown force constants, however, can be reduced through utilization of appropriate symmetry relations of the structure (16). In general, the force constant matrix, [+I, is invariant under point symmetry operation, [S], which leaves the interaction between a pair of atoms undisturbed. Since the interactions in LC model of Fig. 5 are invariant under any rotation about the chain axis, the invariance condition [71

[+I

=

r,

[SI[+IISIT

reduces the force constant,

4, for each

interaction to

where z is taken to be the direction of the chain axis and x is a direction perpendicular to the chain axis. In the development of expressions for Young's moduli only longitudinal motions and therefore the force constants +z, will enter the calculations. , will only enter the expressions for shear Force constants 4 moduli. The derivation of the expressions for shear moduli will be similar to that of Young's moduli but will not be developed here owing to lack of experimental data for comparison. We

for 0

=S xA =S

0.5

TABLE1 . The force-constant parameters utilized in calculations of composition-dependent elastic constants of binary metallic glasses

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

for 0.5 s xA s 1, where W, and W, are defined as

ZrCu ZrNi NbNi TaNi

2.4 1.8 1.1 1.6

0.410 0.265 0.750 0.800

8.6 13.0 13.0 13.0

TABLE2. Crystalline nearest-neighbor interatomic separations utilized in calculations of the mechanical and structural properties of metallic glasses (26) Element

Nearest-neighbor separation (A)

and

Young's modulus for pure A-type material, E,, is given by

m, is . the linear density of the A-type atom. where p, = a At this point, the LC model equations for metal-metal Young's moduli are dependent on three unknowns, namely R , , R,, and E,. The available experimental data on composition-dependent elastic constants of binary metals include ZrCu (17-21), NbNi (20,21), ZrNi (21), and TaNi (21). With the three adjustable parameters of the LC model in [lo] and [l I], the process of comparing the calculations with the experimental data is not totally unique. Although the future availability of a larger number of experimental data points can minimize or even remove this problem, with the existing level of experimental data we need to impose additional restrictions and further limit the number of adjustable parameters. The approach adopted is to limit the range and variation of each of the adjustable parameters through utilization of the interrelationships among the four metal-metal glasses under study. For example, Ni is common to NbNi, TaNi, and ZrNi glasses. Therefore, E(100% Ni) should be the same for all three glasses. Similarly, Zr is common between ZrNi and ZrCu and therefore both glasses should have identical E(100% Zr). ZrCu was chosen as the starting point for comparing theory with experiment since it has the largest number of experimental data. This fixed the value of E(100% Zr), which was later used in ZrNi. Similarly, E(100% Ni) was determined from the calculations of NbNi. This resulted in two-parameter calculations for TaNi and, effectively, a one-parameter calculation for ZrNi. Figures 7 and 8 provide a comparison of the calculations with the experimental data and Table l summarizes the force-constant parameters. The nearest-neighbor intera-

tomic separations utilized in the calculations are given in Table 2. The qualitative and quantitative agreement of the calculations with the experimental data is observed to be good, especially in light of the simplicity of the LC model and the adjustable parameter limitations that effectively reduce the comparison of theory with experiment to a two-parameter problem. The comparison of theory with experiment can be improved by the introduction of next-nearest-neighbor force constants into the model. This option was not utilized this time because of the limited nature of the data available and the new unknown parameters that the model would acquire through extension of the interatomic interaction range. 4. Calculation of structural properties In a one-dimensional system the geometrical factor 4.rrr2 dr of a three-dimensional system changes to d r and hence the expressions for the radial-distribution function g(r) can be defined as (22)

where g&r) is the number of j-type atoms at the end point of vector r when an i-type atom is located at the origin, averaged over all i-type atoms in the structure. g&r) can be defined as (23)

where the prime over the summation excludes the term j = i and < ... > indicates the averaging over all i-type atoms. It must be emphasized that the assumption of the periodicity of the linear chain considerably simplifies the averaging process. The weighing factor w Q is defined as (23)

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

28

CAN. J.

PHYS. VOL.

Composition x

68. 1990

Composition x

Composition x Composition x

FIG.7. Comparison of the calculated (solid line) and experimental Young's moduli of (a) ZrCu (0, Ref. 21; A , ref. 20; 0 , ref. 18; and a, refs. 17 and ref. 19), and (b) ZrNi (0, ref. 21).

FIG. 8. Comparison of the calculated (solid line) and experimental Young's moduli of (a) NbNi (0, ref. 21; A , ref. 20) and (b) TaNi (0, ref. 21).

where

In performing the calculations we will replace the 8 function of [16] by a Gaussian-distribution function of the form 1/2 exp - u (r - rJ2

[18]

k.=

fi -

' fe f.is the atomic scattering factor of the i-type atom, ci the composition, and defined as

fe

the average scattering factor per electron ci f ,

where Zi is the atomic number of the i-type atom. The exact value of the scattering factor f . is dependent on the scattering wave number. Since our calculations are largely concerned with the qualitative features of the model, and for the sake of simplicity of the calculations, we will utilize the following approximation [20]

fi = z,

Equation [20] is strictly valid only at small scattering wavenumber values but will suffice for the purpose of our model calculations.

primarily to account for the finite resolution and broadening of the experimental data and secondly to partially correct, in an average way, for the thermal and disordering displacements around each atom. The value of u is taken equal to 20 in the calculations. Our first calculations utilized the simple LC models given in Figs. 5 and 6. It was observed that the models are capable of distinguishing the different structural characteristics of metallic glasses and of providing good qualitative agreement with the experimental data. For metal-metal glasses (see Fig. 9) the second peak splittings are centered in a rather wide band of compositions around 50% composition. For metal-metalloid glasses (see Fig. 10) the second peak splittings are narrowly centered around the 25% composition of the smaller metalloid atom. If one associates the degree of splitting of the peak with the ease of experimental production of glasses, then the model calculations conform to the experimentally observed

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

RARZADEH

29

FIG. 9. The composition-dependent radial-distribution function of metal-metal glasses calculated using the linear-chain model of Fig. 5.

FIG. 10. The composition-dependent radial-distribution function of metal-metalloid glasses calculated using the linear-chain model of Fig. 6.

characteristic that metal-metalloid glasses are produced around 20% metalloid composition while metal-metal glasses are produced over a wide band of midrange compositions. In Fig. 11, the calculation of the coordination numbers using the LC model of Fig. 6 shows good agreement with the threedimensional computer modeling calculations (24). The LC model and the three-dimensional computer model calculations are normalized to the coordination number of the majority transition metal atom, M, surrounding the minority metalloid atom, m, i.e., N L , ., The observation that LC models as simple as those of Figs. 5 and 6 can distinguish such markedly different structural features is intriguing. However, to obtain qualitative agreement with the details of the experimental data, we need to rely on LC models with more complex building blocks. The LC model with a one-dimensional building block becomes extremely simplistic for monatomic systems such as pure glassy metals. Therefore, the LC model of Fig. 3 with two-dimensional Penrose building block was utilized in the

calculations of pure glassy metal radial-distribution functions. This model has a single adjustable parameter, the angle 0. The experimental data on the structural properties of pure glassy metals are limited. For our study, we utilized the measurements of cobalt's radial-distribution function (25). The ne!rest-neighbor separation was taken as 2.497 (26) (1A = lo-'' m). The comparison of the experimental data with the calculated radial-distribution function is given in Fig. 12 for 0 = 78". The theoretical calcdlations produce the splitting of the second peak and the agreement in intensity and peak position seems to be satisfactory even for the third peak. The next step is to apply the LC model with the Penrose building block to the binary glasses. For the metal-metalloid glasses, Fe,,P,, and Fe,,P,, were chosen for the study. The LC models utilized in the calculations are shown in Fig. 13 for compositions that contain 17% and 25% phosphorus. Using the ne~est-neighbordistances of r , = 2.482 A (26) and r, = 2.56 A (27), Fig. 14 compares the LC model calculations with the experimental data (28,29). The calculations

CAN. J. PHYS. VOL. 68. 1990

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

0

I

Computer Modeling

x (at. % metalloid)

FIG. 1I . Comparison of the calculated coordination number of metal-metalloid glasses using the LC model of Fig. 6 with the results of computer modeling (24). TABLE3. The model parameters for the transition metal - transition metal radial-distribution functions. The angles of planar close packing for corresponding crystalline structures are also given for comparison

Linear Chain Model Experiment

Metal A

........... ....

A,

-,*I

4

2 r

6

8

(1)

FIG. 12. Comparison of the LC model calculations based on Fig. 3 with the experimental observations on amorphous cobalt (25).

are based on a single angle, 8, for all Penrose units, thus creating a one-parameter model. The calculations for Fig. 14 correspond to the angle 0 = 92.5". The results show good agreement with the experimental data in both peak positions and relative peak intensities. The second peak splittings show the correct ratio of the subpeaks, which is in agreement with experimental results. For Fe8,P1,, the agreement is extended to the third and fourth peaks of the

0

zr70Ni30

66" 72" 72"

zr7oc0,o

72"

Zr70Pd30 Zr,oFe30

0

Metal B

Metallic glass 00 60" 60" 60" 60"

0 103" 80" 106" 68"

00 90" 70.53" 90" 60"

g

0.10 0.13 0.16 0.13

observed data. The study of partial radial-distribution functions of Fe,,P,, and Fe8,P1, clearly shows that the metal-metal partial radial-distribution function predominantly determines the qualitative and quantitative features of the total radial-distribution function. Our last study of binary glassy systems considered Zr,, Fe,,, Zr,, Co,,, Zr,, Ni,,, and Zr,, Pd,, metallic glasses (30). The LC model utilized in the calculations is shown in Fig. 15. For transition metal glasses, the Zr-Zr, Zr-TM, and TM-TM nearest-neighbor distances are smaller than the values estimated from the Goldschmidt radii (3'1). Therefore, in our calculations, we defined the nearest-neighbor separations by r = ro(l-g), where r, is the Goldschmidt radius (26) and g is a parameter to be determined from the experimental data. Table 3 summarizes the parameter values for 8 and g utilized in the calculation of transition metal radial-distribution functions. The calculations are compared with the experimental data in Fig. 16 and produce the peak positions, the observed splitting of the second peak, and even the fine structure of the first peak. The agreement is judged to be excellent for such a simple model of the glassy system.

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

RAFIZADEH

FIG. 13. Linear-chain models of (a) Fe,,P,,

FIG. 14. Comparison of the LC model calculations (-) for (a) Fe,,P,,, (28) and (b) Fe,,P,, (29).

and (b) Fe,,P,, utilized for calculation of the radial distribution functions.

of the radial distribution function with the experimental observations (dashed lines)

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

CAN. J. PHYS. VOL. 68, 1990

FIG. 15. The linear-chain model of transition-metal - transition-metal glasses utilized in the calculations of radial-distribution functions.

FIG. 16. Comparison of the linear-chain model calculations (-)

with the experimental data (---) for transition metal glasses (30).

1

33

RAFIZADEH

Can. J. Phys. Downloaded from www.nrcresearchpress.com by MICHIGAN STATE UNIV on 01/28/17 For personal use only.

Overall, the LC models of the metallic glasses are observed to be quite successful in calculations of the structural properties such as radial-distribution functions. Considering the ease of LC model calculations, these simple models are not only useful by themselves, but can also be quite valuable in planning the more costly and time-consuming experiments. The LC models are not restricted to binary systems or Penrose building blocks and can readily incorporate many-component glasses and three-dimensional building blocks into the analysis, if so desired.

In conclusion, we believe that the linear-chain model of a glassy system is a successful proposition. It provides a simple model of metallic glasses, especially when multicomponent systems are being considered. The model can simultaneously account for the mechanical and the structural properties in the current study and can easily be applied to new areas such as thermal or other properties of the glassy materials.

Acknowledgement

5. Discussion and conclusions

Part of this study was completed at the Institute of Chemical Analysis, Applications and Forensic Science, Northeastern University, Boston.

In this study, a linear-chain model of glassy metals is proposed. Different paradigms with one- and two-dimensional building blocks are successfully utilized in the analysis of composition-dependent elastic constants and radial-distribution functions. Notwithstanding the simplicity of the LC models, the comparison of the calculations with the experimental data reveals both qualitative and quantitative agreements over a relatively wide range of physical properties. One of the assumptions utilized in LC models is that the broken translational symmetry of the glassy structure can be approximated with long-range translational symmetry. The results of our calculations demonstrate that this approximation does not affect the utility of the LC model as a simple analytical tool. On the other hand, the LC model calculations of the elastic constants show that the significant characteristic determining the composition dependence of the elastic constants is an inversion asymmetry of the atomic positions. In metallic glasses, no atom is located at an inversion center and this characteristic is largely replicated in the LC models. From the results of our calculations it seems that the inversion asymmetry could be a more significant modeling requirement as compared with the broken translational symmetry. From a different perspective, one can revert back to Bernal's original study of the hard sphere random systems (8) to revive the significance of a different aspect of Bernal's observations. Bernal's study of the physical hard sphere random systems highlighted two distinct features of the structure, "holes" and "collineations. " Following Bernal's physical experiments, the concept of the hole as a structural unit of glassy structure has been adopted in a variety of modeling activities. The collineations, however, were not pursued beyond Bernal's initial observations. Using Bernal's definition, collineations consist of a "long string of molecules in a more or less straight line." Bernal's observations, in our opinion, allow modeling of the glassy system using any of the following concepts: (i) a structure consisting of holes as building blocks; (ii) a structure consisting of collineations as building blocks; (iii) a combination of (i) and (ii) above. Our approach utilizing a linear-chain model with onedimensional building blocks seems to fall into category 2, while the linear-chain model with Penrose holes falls into category 3 where the linear collineation-type structure (Penrose units on a straight line) is combined with the holelike characteristics of the Penrose unit. Our results seem to support the initial Bernal observations that holes and collineations are both characteristics of the glassy structure, and thus, in our opinion, could equally account for the physical properties in the modeling of the metallic glasses.

1. P. H. GASKELL. In Glassy metals 11. Edited by H. Beck and H.-J. Guntherodt. Springer-Verlag, Berlin. 1983. p. 5. 2. U. GONSER, J. Non-Cryst. Solids, 61 & 62, 1419 (1984). 3. J. L. FINNEY. Nature (London), 266, 309 (1977). and H. J. FROST.Phys. Rev. B, 23, 1506 4. D. S. BOUDREAUX (1981). 5. D. WEAIRE, N. F. ASHBY,J. LOGAN, and M. J. WEINS.Acta Metall. 19, 779 (1971). 6. F. CYROT-LACHMAN. Phys Rev. B, 22, 2744 (1980). 7. P. DEAN.Rev. Mod. Phys. 44, 127 (1972). 8. J. D. BERNAL. Proc. Roy. Soc. London A, 280, 299 (1964). Sci. Am. 236, 110 (1977). 9. M. GARDNER. 10. D. E. POLK.Scr. Metall. 4, 117 (1977). 11. J. F. SADOC and R. MOSSERI. J. Non-Cryst. Solids, 61 & 62, 487 (1984). 12. D. R. NELSON. J. Non-Cryst. Solids, 61 & 62, 475 (1984). 13. J. HAFNER. In Glassy metals I. Edited by H.-J. Guntherodt and H. Beck. Springer-Verlag, Berlin. 1981. p. 93. 14. H. S. CHEN,J. T. KRAUSE, and E. COLEMAN. J. Non-Cryst. Solids, 18, 157 (1975). 15. H. A. RAFIZADEH. Can. J. Phys. 68, 14, (1990). 16. S. BHAGAVANTAM. Crystal symmetry and physical properties. Academic Press, London. 1966. 17. H. S. CHEN.Mater. Sci. Eng. 25, 59 (1976). 18. L. A. DAVIS,C.-P. CHOU,L. E. TANNER, and R. RAY.Scr. Metall. 10, 937 (1976). 19. T. MASUMOTO and R. MADDIN. Mater. Si. Eng. 19, 1 (1975). 20. 11. S. CHENand J. T. KRAUSE. Scr. Metall. 11, 761 (1977). 21. S. H. WHANG,H. HWANG, and B. C. GIESSEN. Presented at Material Research Society Annual Meeting. Boston. 1979. 22. R. KAPLOW, S. L. STRONG, and B. L. AVERBACH. In. Local atomic arrangements studied by x-ray diffraction. Edited by J. B. Cohen and J. E. Hillard. Gordon and Breach. New York 1966, p. 159. 23. C. N. J. WAGNER. In Liquid metals. Edited by S. Z. Beer. Marcel and Dekker, New York 1972. p. 257. 24. D. S. BOUDREAUX. Phys Rev. B, 18, 4039 (1978). 25. P. K. LEUNG and J. G. WRIGHT. Philos. Mag. 30, 185 (1974). 26. T. LYMAN. (Editor). Metals handbook. Vol. 1. 8th ed. American Society of Metals, Melno Park, OH. 1961. 27. B. WUNDERLICH. Macromolecular physics. Vol. 1. Academic Press, New York. p. 44. 28. Y. WASEDA, H. OKAZAKI, and T. MASUMOTO, ROC.Int. Conf. Struct. Non-Cryst. Mater. Taylor and Francis, London. 1977. p. 95. 29. Y. WASEDA, H. OKAZAKI, and T. MASUMOTO. ROC.Int. Conf. Struct. Non-Cryst. Mater. Taylor and Francis, London, 1977. p. 202. 30. Y. WASEDA and H. S. CHEN.In Rapidly quenched metals 111. Vol. 1. Edited by B. Cantor. Metals Society, London. 1978. p. 415. 31. H. S. CHENand Y. WASEDA.Phys Status Solidi A, 51, 593 (1979).