Rigidity percolation in glassy structures

Journal of Non-Crystalline Solids 76 (1985) 109-116 North-Holland, Amsterdam

109

Section 1II. Modeling RIGIDITY P E R C O L A T I O N IN GLASSY S T R U C T U R E S M.F. T H O R P E Department of Physics and Astronomy, Michigan State Universi(v, East Lansing, M1 48824. USA

We review the new concept of rigidiO"percolation and show that if local flexible units are joined together to form a network, the composite consists of floppy and rigid regions. When the rigid regions percolate, the whole network becomes rigid and resists attempts at elastic deformation. These ideas are applied to network glasses. It is shown that in the floppy region there exist low-frequencv modes that should show up in inelastic neutron scattering.

1. Introduction In this talk we review some recent new concepts which can be grouped together under the general heading rigidity percolation. These ideas find their most important application in network glasses and we discuss an inelastic neutron scattering experiment that could help to clarify our ideas in this area. Most of this work has been published elsewhere and the original references are given in the text wherever relevant. The suggestion of observing the floppy modes using inelastic neutron scattering is new. Percolation theory has m a n y applications in physics [1]. A simple example is the magnetism of an insulating alloy like R b M n , M g 1 , F 4 where Mn is magnetic and interacts with its nearest neighbors. W h e n Mg, which is n o n m a g netic, is substituted for Mn, the N6el temperature drops. The N6el temperature separates the ordered antiferromagnetic phase from the disordered paramagnetic phase. Eventually, as p decreases, the N6el temperature goes to zero at p~, which is when the material breaks up into isolated magnetic Mn islands each surrounded by nonmagnetic Mg. The quantity Pc is k n o w n as the percolation concentration; p ~ - 0.31 for site percolation on a simple cubic lattice. M a n y other physical properties have been studied through the percolation transition. A classic example [1-3] is the resistance of a network of wires (bonds) that are joined at nodes (sites). The network is depleted by removing b o n d s or sites until it ceases to conduct when the conductivity goes to zero. This occurs when the network geometrically separates so that there are no paths for the current to take. [1-3] We refer to this general class of problems as connectivity percolation. The conductivity o of a network goes to zero at the percolation concentration Pc as

o-(p-pc

)'

0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(1)

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M. F Thorpe / Rigidity percolation in glassy structures

for small ( p -p~). For two-dimensional networks t - 1.3 while for three-dimensional networks t - 1.8 [4]. Surprisingly, the elastic properties of such networks have not been considered until quite recently. These are much more interesting because the variety of possible behavior is much richer. The elastic properties of a system are described by a fourth-rank tensor, which has two independent components even in an isotropic system. The electrical properties of a similar system are described by a second-rank tensor which only has one independent component in a system with cubic or higher symmetry. From work done in the last couple of years it appears that the elastic constants C,~ go to zero at p~ as

C,,- (p-pc)/,

(2)

where f - 3 . 5 in 2D [5-7]. The underlying geometry is that of connectivity percolation and eq. (2) serves to define a new exponent f. The reason that f is larger than t can be understood on the "nodes and links" picture [6]. Until quite recently it was thought that f = t [8]. We refer to those problems involving connectivity percolation as class 1 problems. A recent experiment in which holes were punched in thin metal sheets is consistent with t - 1.4 and f - 3.5 [71.

2. Rigidity percolation A much more interesting class of problems involves rigidity percolation. We shall refer to these as class-2 problems. It is simplest to illustrate this with an example. Imagine a triangular net of mass points connected by nearest-neighbor central forces. The system is stable and elastically isotropic so there are two independent elastic constants Ctl and C44. If bonds are randomly removed with probability (1 - p ) , then both Cll and C44 decrease as p decreases from 1 as shown in fig. 1. Connectivity percolation occurs at Pc = 2 sin(~r/18)= 0.35 [9], but the elastic constants vanish around p* = _~. Thus for 0.35 _

(3)

M.F. Thorpe / Rigidity percolation in glassy structures

1.4

111

Triangular Net /

1.2 •- 1.0 "o o 0.8

VCll

at/

_ o C/44/ Oi

".,= 0.6

0

0.2

0.4

0.6

~__

p

' " 0.4

0.81 ~

0.2 0

0

I 0.1

1 I I I 0.2 0.3 0.4 0.5 P

0.6

0.7 0.8 0.9

Fig. 1. The elastic constants Cll and C,~ averaged over three configurations for a 440-atom triangular network. The insert shows the fraction of zero-frequency modes f for a 168-atom triangular network averaged over three configurations. The straight lines are from the effectivemedium theory described in the text (from ref. [10]).

Fig. 2. A two-coordinated bridge connecting two regions. The bridge is ineffective in transmitting any elastic restoring force and can be trimmed (from ref. [10]).

"1" Polymeric Glass

TF Amorphous Solid

Fig. 3. The Rigid and Floppy regions in networks. Type I (polymeric glass) has zero elastic constants whereas type II (amorphous solid) has finite elastic constants (from ref. [11]).

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M.F. Thorpe / Rigid#)' percolation in glassy structures

and if b > 2 j - 3 the network is rigid while if b < 2 j - 3 the network is floppy. The equality (3) is the marginal case. This can be proved by building up a network of triangles. A triangle is the smallest rigid unit. For one triangle b = 3, j = 3. Each time another triangle is added, Ab = 2 and A j = 1, so that eq. (3) is proved by induction. Closer examination shows that eq. (3) is only approximately true and can be violated. However it is remarkably accurate for most homogeneous systems. Similar expressions to eq. (3) can be derived for 3D networks. It is useful to reinterpret eq. (3). The number of degrees of freedom of the network is 2 j and the number of constraints is just the number of bars b. This leads to a quantity F which is the number of independent macroscopic deformations. Clearly F= 2j-

b - 3,

(4)

where the 3 represents the 2 translations and 1 rotation of the whole system. So the system is unstable if F > 0. These ideas can be applied directly to the triangular network with missing bonds that we have already discussed. If there are N sites, then there are d N degrees of freedom (where d = 2 in the present case). The number of bonds present is z N p / 2 (where z = 6 is the number of neighbors). If F = f d N then F = dU - z N p / 2 ,

(5)

f= 1 - zp/2d

(6)

or is the fraction of zero-frequency modes. These correspond to deformations that cost no energy. We see that f goes to zero at (7)

p* = 2 d / z ,

which for the triangular net gives p* = 2 in close agreement with the numerical simulations shown in fig. 1. Showing that there is nothing special about the triangular net, similar results have been obtained in 3D for the fcc lattice [10] where p* - ½. We have computed f numerically for the triangular net [10]. The dynamical matrix was set up and diagonalized numerically and the number of zerofrequency eigenvalues counted. The results are shown as the insert in fig. 1. They are in good agreement with the prediction of eq. (6) using the constraints counting method. We have developed an effective-medium theory for the elastic constants [10]. The result is that

Cl) - -

c o,

C44 -

P -p* -

c°

l-p*'

(8)

where C °] and C° are the elastic constants at p = 1. The numerical simulations are remarkably close to these straight lines as can be seen in fig. 1. There presumably are deviations from mean-field behavior close to p*, but they are exceedingly small. Equally good results are obtained for the fcc lattice [10] and

M.F. Thorpe / Rigidity percolation in glassy structures

113

also for the square net with first- and second-neigbor central forces [13]. These systems are described more accurately by effective-medium theory than any other system we know of. The reasons are unclear but it is important to understand this agreement better. Similar effective-medium theories for the conductivity of resistor networks do not work as well [1-3].

3. Application to glassy materials This work finds its most important application in non-crystalline materials. Some years ago J.C. Phillips discussed underconstrained and overconstrained glasses [14]. These are the same as our rigid and floppy regions. Imagine that a particular r a n d o m network with N atoms has been constructed with n r atoms having r bonds (r = 2, 3 and 4):

N = Y2n,.

(9)

r

The largest forces in covalent networks are the nearest-neighbor bond-stretching force (force constant a) and the angle-bending force (force constant fl). The local floppiness is now not caused by pin joints but in 3D by the indeterminate dihedral angle. There are 3N degrees of freedom. There is one constraint associated with each b o n d (giving r/2 per atom) and 2r - 3 angular constraints associated with each r-coordinated atom. The n u m b e r of zerofrequency modes is

3N - Y'.nr[r/2 + (2r - 3)],

(10)

r

a n d ' t h e fraction of zero-frequency modes f is f= 2-~(r),

(11)

where the mean coordination ( r ) is defined by

(r) = E r n J E n r . r

(12)

r

The n u m b e r of zero-frequency modes goes to zero at ( r ) = re = 2.4. To illustrate this, we have r a n d o m l y removed bonds from a d i a m o n d lattice, thereby creating sites that are 2, 3- and 4-coordinated. Both the elastic constants and the fraction f were calculated for these networks and the results are shown in fig. 4. It can be seen that there is indeed a transition around re = 2.4 as first suggested by Phillips. We do not at present have an effective-medium theory for the C,/ as it is difficult to include the angular forces in a satisfactory way. Thus we see that r a n d o m networks may be separated into two kinds [11]: Type 1: Polymeric glasses. If ( r ) < re as in, say, GeeSe 1 , with x < 0.2, then the picture is of polymer chains (Se) with some cross links (Ge). The elastic constants are zero and the system is mechanically unstable. In reality, the small

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M.F. Thorpe / Rigidity percolation in glassy structures

,

I

I 0.3

1.5

J

[] 0.2

8

f

1.0 Cll

R

0.1 a 0.5 2

,,A^J,^^ 0

2.2<:~>4

0 2

2.4

2.6

~

I 2.8

3.2

I 3.6

4

Fig. 4. The elastic modulus C N with fl/a = 0.2 as a function of mean coordination ( r ) for three different random networks with 516 atoms each. The insert shows the number of zero-frequency modes f, averaged over three 216-atom networks, compared to the result of eq. (11) (from ref.

[15]). dihedral angle and Van der Waals forces will stabilize the structure and determine the elastic properties. These small forces play an essential role in stabilizing the structure. Type H: Amorphous solids: If ( r ) > r: as in, say, Ge, Se 1_, with x > 0.2, then we have a "tight" network that is mechanically stable and has elastic constants determined by the large a, /3 covalent forces. The weak dihedral angle and Van der Waals forces are not important and can be ignored for most purposes. We have used the words polymeric glass and amorphous solid just so as to have convenient labels to talk about. However, it is interesting that ( r ) = r: = 2.4 does roughly divide the bulk glasses that can be quenched from the melt from amorphous solids that can only be made in thin films.

4. An inelastic neutron scattering experiment It can be seen from fig. 4 that in the polymeric glass ( ( r ) < r:) phase, there are a large number of zero-frequency modes. For ( r ) = 2, one third of the modes ( f = -~) are at zero frequency. This number decreases, roughly linearly, and goes to zero around ( r ) = re = 2.4 when the transition to the amorphous solid takes place. This result is based on the inclusion of only the two large covalent forces, the nearest-neighbor bond-stretching and bond-bending forces

M.F. Thorpe / Rigidity percolation in glassy structures

115

c~, ft. In reality, it is important to include the weaker further-neighbor forces in order to stabilize the structure of the polymeric glass. Such forces include the dihedral-angle force and the even longer-ranged Van der Waals forces. We will collectively assign these forces a force constant 7 where y << et, ft. The 7 forces are also present, of course, in the amorphous solid, but do not lead to any new qualitative effects. A p a r t from leading to structural stability in the polymeric glass, the y forces move the delta function at 0~2 = 0 up to a frequency t 0 2 - 7 / M where we would expect a fairly well-defined low-frequency band. Here M is a typical atomic mass in the network. We will refer to this low-frequency peak as " s p o n g y modes" to indicate that they have a small infinite frequency. The width of the s p o n g y - m o d e peak is u n k n o w n at this time and will need further theoretical investigation. We might suspect that it is reasonably sharp since the structure is rather homogeneous on some reasonable length scale. By examining the large-Q, o n e - p h o n o n inelastic neutron scattering for a series of c o m p o u n d s like, say, GeeSe ~_,, it should be possible to study the evolution of the s p o n g y - m o d e peak. Primary interest is focused on the weight in the peak that should track f in fig. 4. The position of the peak is of secondary interest.

5. Conclusion There is very little direct experimental evidence from glasses to support the floppy-rigid picture presented here, although we believe it to be correct. We have pointed out that experiments with pulsed neutron sources could clarify the situation and help in our understanding. I would like to thank my graduate students, E. Garboczi and H. He, who did much of the work described here. Full details can be found in the references cited. I would also like to thank the O N R (contract N00014-80-C0610) and the N S F (grant D M R 8317610) for financial support.

References [1] See for example S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [2] R. Zallen, The Physics of Amorphous Solids (Wiley, New York, 1983). [3] J.P. Straley, Percolation Structures and Processes, in: Ann. Israeli Phys. Soc., Vol. 5, eds. G. Deutscher, R. Zallen and J. Adler (1983). [4] S. Feng and P.N. Sen, Phys. Rev. Lett. 52 (1984) 216. [5] D.J. Bergman, Phys. Rev. 31 (1985) 1696. [6] Y. Kantor and I. Webman, Phys. Rev. Lett. 52 (1984) 1891. [7] L. Benguigui, Phys. Rev. Lett. 53 (1984) 2028. [8] P.G. de Gennes, J. de Phys. 37 (1976) L-1. [9] J.W. Essam, in: Phase Transitions and Critical Phenomena, Vol. 2, eds. C. Dombandi and M.S. Green (Academic Press, London, New York) p. 197.

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M.F. Thorpe / Rigidi(v percolation in glas:~v structures

[10] S. Feng, M.F. Thorpe and E. Garboczi, Phys. Rev. B31 (1985) 276. [11] M.F. Thorpe, J. Non-Cryst. Solids 57 (1984) 250. [12] J.C. Maxwell, Phil. Mag. 27 (1864) 294. See also A.J.S. Pippard and J.F. Barker, The Analysis of Engineering Structures (Arnold, London, 1943). [13] E.J. Garboczi and M.F. Thorpe, Phys. Rev. B31 (1985) 7276. [14] J.C. Phillips, J. Non-Cryst. Solids 34 (1979) 153; Phys. Stat. Sol. (b) 101 (1980) 472 and Phys. Today (Feb. 1982) 1. [15] H. He and M.F. Thorpe, Phys. Rev. Lett. 54 (1985) 2107.