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0021-3640/04/7912-$26.00 © 2004 MAIK “Nauka/Interperiodica”

0632

JETP Letters, Vol. 79, No. 12, 2004, pp. 632–634. Translated from Pis’ma v Zhurnal Éksperimental’no

Original Russian Text Copyright © 2004 by Ojovan.

œ

i Teoretichesko

œ

Fiziki, Vol. 79, No. 12, 2004, pp. 769–771.

Percolation transitions attract considerable interest,

because they offer an explanation for a wide class of

phenomena [1–3]. For example, the glass transition in

spin glasses is explained on the basis of the percolation

theory [4]. At the same time, the nature of glass transi-

tion in oxide systems is not yet clearly understood [5–

8]. Amorphous SiO

2

, as the simplest glass-forming

material, is suitable for use in the model studies in this

area of research. At temperatures higher than

amorphous SiO

2

transforms to the supercooled liquid

state, whereas, below

T

g

, it is in the glassy solid state.

The changes occurring in the atomic system as the tem-

perature passes through

T

g

gated. According to the concept proposed by Hunt, the

material at temperatures above

transport regime, while at low temperatures, it is in the

diffusive transport regime [5]. Major progress in the

understanding of the structural changes of an amor-

phous material passing through

the help of the molecular dynamics (MD) modeling [6]

and, in particular, by studying the Voronoi polyhedra

(analogues of the Vigner–Seitz cell) [7, 8]. The MD

experiments showed that, in the liquid state, percolation

clusters composed of Voronoi coordination polyhedra

with low-density atomic configurations are formed in

the material, while no such clusters occur in the solid

(glassy) state [7]. However, in the solid state, percola-

tion clusters of Voronoi coordination polyhedra with

high-density (compact) atomic configurations are

T

g

= 1475 K,

have been much investi-

T

g

is in the percolative

T

g

was achieved with

formed [7, 8]. Since the percolation clusters of Voronoi

coordination polyhedra with low-density atomic con-

figurations exist in the liquid state only, it is possible to

distinguish between the liquid and solid (glassy) states

of amorphous materials on the basis of the MD experi-

ments [7]. At the same time, the MD experiments show

that, near the glass-transition temperature, the geome-

try of an amorphous material changes because of the

formation of the fractal percolation clusters [2].

This paper shows that, as the temperature passes

through

T

g

in amorphous SiO

occurs in the system of the network defects presumably

consisting of defective SiO molecules. The transition

can be traced analytically, making it possible to derive

a simple expression for the glass-transition tempera-

ture. The analytic calculation is based on the Doremus

viscosity model (D model) relating the viscosity of the

amorphous material to the thermodynamic parameters

of the network defects [9–11].

2

, a percolation transition

An amorphous material can be represented by a

topologically disordered network. The three-dimen-

sional network of amorphous SiO

rahedra bridged by oxygen atoms. A perfect network of

an amorphous material has no defects at absolute zero,

but defects arise at finite temperatures

of defects depends on the Gibbs free energy of a defect:

2

consists of SiO

4

tet-

T

. The formation

(1)

Gd

Hd

TSd,–=

Glass Formation in Amorphous SiO

Transition in a System of Network Defects

2

as a Percolation Phase

M. I. Ojovan

Sir Robert Hadfield Building, University of Sheffield, Sheffield, S1 3JD, United Kingdom

Received May 6, 2004

Thermodynamic parameters of defects (presumably, defective SiO molecules) in the network of amorphous

SiO

2

are obtained by analyzing the viscosity of the melt with the use of the Doremus model. The best agreement

between the experimental data on viscosity and the calculations is achieved when the enthalpy and entropy of

the defect formation in the amorphous SiO

2

network are

analysis of the network defect concentration shows that, above the glass-transition temperature (

form dynamic percolation clusters. This result agrees well with the results of molecular dynamics modeling,

which means that the glass transition in amorphous SiO

Below

T

g

, the geometry of the distribution of network defects is Euclidean and has a dimension

the glass-transition temperature, the geometry of the network defect distribution is non-Euclidean and has a

fractal dimension of

d

f

= 2.5. The temperature

T

g

can be calculated from the condition that percolation arises in

the defect system. This approach leads to a simple analytic formula for the glass-transition temperature:

H

d

/(

S

d

+ 1.735

R

). The calculated value of the glass-transition temperature (1482 K) agrees well with that

obtained from the recent measurements of

T

g

for amorphous SiO

odica”.

PACS numbers: 61.43.Dq; 64.60.Ak; 64.70.Pf; 66.20.+d

H

d

= 220 kJ/mol and

S

d

= 16.13

R

, respectively. The

T

g

), the defects

2

can be considered as a percolation phase transition.

d

= 3. Above

T

g

=

2

(1475 K).

© 2004 MAIK “Nauka/Interperi-

Page 2

JETP LETTERS

Vol. 79

No. 12

2004

GLASS FORMATION IN AMORPHOUS SiO

2

633

where

tion of one mole of defects. Doremus assumed that the

diffusion and viscous flow in silicates proceed through

the formation of defective SiO molecules. The forma-

tion of these defects favors the appearance of five-coor-

dinate Si and O atoms, which was confirmed experi-

mentally in [9]. The formation of defects in the network

of amorphous SiO

2

can be represented by the reaction

H

d

is the enthalpy and

S

d

is the entropy of forma-

(2)

where (–Si–)

(–Si–)

defect

defects. Let the concentration of the elementary blocks

of the network be

C

0

and the defect concentration be

[(

−

Si–)

defect

] = [(–O–Si–)

defect

[(–O–Si–)

net

] = (

C

0

–

C

d

). The equilibrium reaction con-

stant for (2) depends on the change in the Gibbs energy

G

= 2

G

d

:

net

and (–O–Si–)

and (–O–Si–)

net

refer to the network and

are the bond-rupture

defect

] =

C

d

. Then, [(–Si–)

net

] =

(3)

Hence, the equilibrium content of defects is determined

as (see also [11, 12])

(4)

To calculate the concentration of network defects in

amorphous SiO

2

, it is necessary to know the numerical

values of the enthalpy

H

d

and entropy

mation. Both these quantities,

the expression for the viscosity in the D model [10, 11]:

S

, are involved in

d

of defect for-

H

d

and

S

d

(5)

where

radius,

symmetry parameter,

frequency, and

of defect motion. By processing the experimental data

on viscosity, it is possible to obtain the exact values of

H

d

and

S

d

. The results of this analysis are shown in

Fig. 1, which displays the viscosity of amorphous SiO

calculated from Eq. (5) and the experimental data on

viscosity from [13, 14]. The best agreement between

the viscosity calculated from Eq. (5) and the experi-

mental data [13, 14] is achieved with

and

S

d

= 16.13

R

, where

R

is the universal gas constant.

Note that the value

H

d

= 220 kJ/mol is practically equal

to half the strength of one bond for Si in SiO

(443 kJ/mol [15]), which agrees with the physical

meaning of this quantity.

k

D

is the Boltzmann constant,

=

f

αλν

,

f

is the correlation factor,

λ

is the hopping distance,

S

m

and

H

m

are the entropy and enthalpy

r

is the defect

α

0

2

is the

ν

is the

2

H

d

= 220 kJ/mol

2

–Si–

–Si–

(

()net

)defect

–O–Si–

–O–Si–

(

+

()net

+

)defect,

K

∆G/RT

–

(). exp=

Cd

C0

Gd/RT

Gd/RT

–

(

–

()

exp

+

1

)

exp

------------------------------------------ -.

=

η T

( )

kT

6πrD0

--------------- -

Sm

R

-----–

exp=

×

Hm

RT

-------

1

Sd

R

---- -–

Hd

RT

-------

expexp+, exp

Now, let us consider the evolution of the network

defect concentration in amorphous SiO2 with increas-

ing temperature. The results of calculating the relative

concentration ρ = Cd/C0 of defects by Eq. (4) are shown

in Fig. 2.

The defect clusterization is unlikely as long as the

defect concentration is small. As the defect concentra-

tion increases, the formation of clusters becomes more

and more probable. The relative defect concentration is

a function of temperature, ρ(T) = Cd/C0, and increases

with temperature T. A percolation cluster of network

defects is formed when the relative defect concentra-

tion ρ = Cd/C0 reaches the critical value:

ρ T

( )

=

(6)

ρc.

Fig. 1. Viscosity of amorphous SiO2: the curve is calculated

from Eq. (5), and the experimental data are taken from

[13, 14].

Fig. 2. Concentration of network defects in amorphous

SiO2. Above Tg, the defect geometry becomes fractal with

the dimension df = 2.5.

Page 3

634

JETP LETTERS Vol. 79 No. 12 2004

OJOVAN

For a three-dimensional space, the critical density value

is determined by the Scher–Zallen invariant ϑc = 0.15 ±

0.01 [1, 16]. Hence, one can determine from Eq. (6) the

percolation transition temperature. Taking into account

that ρ(T) in equilibrium can be determined from

Eq. (4), we obtain for the glass-transition temperature

Hd

Sd

R

([

ln+

(7)

At temperatures T < Tg, no percolation clusters occur in

the material and the geometry of the network defects

remains Euclidean (d = 3). When T > Tg, a percolation

cluster is formed with the fractal geometry of dimen-

sion df = 2.5 [2]. The network defects are mobile, and,

hence, the percolation cluster is dynamic in character

(from the viscosity data and from Eq. (5), it follows that

the enthalpy of the network defect motion is Hm =

525 kJ/mol). Dynamic percolation clusters with the

dimension df = 2.6 were experimentally observed in

emulsions [17]. It is also significant that the relaxation

processes near the percolation threshold are nonexpo-

nential and described by the Kohlrausch law [2–5]. At

temperatures T > Tg, amorphous SiO2 is a supercooled

liquid, while below Tg, it transforms to the glassy state.

Formula (7) for amorphous SiO2 yields Tg = 1482 K.

This value is only slightly higher than the known value

of Tg = 1450 K (see, e.g., [6]). However, it virtually

coincides with the recent data of scanning calorimetric

measurements: (Tg)exp = 1475 K [18].

Thus, the glass formation in amorphous SiO2 can be

considered as a percolation transition in the system of

network defects (presumably, defective SiO molecules)

with a change in the geometry of the defects from frac-

tal in the liquid state to Euclidean in the glassy state.

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Translated by E. Golyamina

Tg

1

ϑc

–

)/ϑc

]

---------------------------------------------------- -.

=