0021-3640/04/7912-$26.00 © 2004 MAIK “Nauka/Interperiodica”
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.
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 . At the same time, the nature of glass transi-
tion in oxide systems is not yet clearly understood [5–
8]. Amorphous SiO
, as the simplest glass-forming
material, is suitable for use in the model studies in this
area of research. At temperatures higher than
transforms to the supercooled liquid
state, whereas, below
, it is in the glassy solid state.
The changes occurring in the atomic system as the tem-
perature passes through
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 . Major progress in the
understanding of the structural changes of an amor-
phous material passing through
the help of the molecular dynamics (MD) modeling 
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 . However, in the solid state, percola-
tion clusters of Voronoi coordination polyhedra with
high-density (compact) atomic configurations are
= 1475 K,
have been much investi-
is in the percolative
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 . 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 .
This paper shows that, as the temperature passes
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].
, 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:
consists of SiO
. The formation
Glass Formation in Amorphous SiO
Transition in a System of Network Defects
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
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
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
, 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
= 2.5. The temperature
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:
). The calculated value of the glass-transition temperature (1482 K) agrees well with that
obtained from the recent measurements of
for amorphous SiO
PACS numbers: 61.43.Dq; 64.60.Ak; 64.70.Pf; 66.20.+d
= 220 kJ/mol and
, respectively. The
), the defects
can be considered as a percolation phase transition.
= 3. Above
© 2004 MAIK “Nauka/Interperi-
GLASS FORMATION IN AMORPHOUS SiO
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 . The formation of defects in the network
of amorphous SiO
can be represented by the reaction
is the enthalpy and
is the entropy of forma-
defects. Let the concentration of the elementary blocks
of the network be
and the defect concentration be
] = [(–O–Si–)
] = (
). The equilibrium reaction con-
stant for (2) depends on the change in the Gibbs energy
refer to the network and
are the bond-rupture
. Then, [(–Si–)
Hence, the equilibrium content of defects is determined
as (see also [11, 12])
To calculate the concentration of network defects in
, it is necessary to know the numerical
values of the enthalpy
mation. Both these quantities,
the expression for the viscosity in the D model [10, 11]:
, are involved in
of defect for-
of defect motion. By processing the experimental data
on viscosity, it is possible to obtain the exact values of
. 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
is the universal gas constant.
Note that the value
= 220 kJ/mol is practically equal
to half the strength of one bond for Si in SiO
(443 kJ/mol ), which agrees with the physical
meaning of this quantity.
is the Boltzmann constant,
is the correlation factor,
is the hopping distance,
are the entropy and enthalpy
is the defect
= 220 kJ/mol
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:
Fig. 1. Viscosity of amorphous SiO2: the curve is calculated
from Eq. (5), and the experimental data are taken from
Fig. 2. Concentration of network defects in amorphous
SiO2. Above Tg, the defect geometry becomes fractal with
the dimension df = 2.5.
JETP LETTERS Vol. 79 No. 12 2004
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
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 . 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 . 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., ). However, it virtually
coincides with the recent data of scanning calorimetric
measurements: (Tg)exp = 1475 K .
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.
1. B. I. Shklovskiœ and A. L. Efros, Electronic Properties of
Doped Semiconductors (Nauka, Moscow, 1979;
Springer, New York, 1984).
2. M. B. Isichenko, Rev. Mod. Phys. 64, 961 (1992).
3. M. I. Klinger, Phys. Rep. 165, 275 (1988).
4. I. Campbell, P.-O. Mari, A. Alegria, and J. Colmenero,
Europhys. News 3/4, 46 (1998).
5. A. G. Hunt, J. Non-Cryst. Solids 274, 93 (2000).
6. K. Binder, J. Non-Cryst. Solids 274, 332 (2000).
7. N. N. Medvedev, A. Geider, and W. Brostow, J. Chem.
Phys. 93, 8337 (1990).
8. A. V. Evteev, A. T. Kosilov, and E. V. Levchenko, Pis’ma
Zh. Éksp. Teor. Fiz. 76, 115 (2002) [JETP Lett. 76, 104
9. R. H. Doremus, J. Appl. Phys. 92, 7619 (2002).
10. M. I. Ozhovan, Pis’ma Zh. Éksp. Teor. Fiz. 79, 97 (2004)
[JETP Lett. 79, 85 (2004)].
11. M. I. Ojovan and W. E. Lee, J. Appl. Phys. 95, 3803
12. J. W. Haus and K. W. Kehr, Phys. Rep. 150, 263 (1987).
13. G. Urbain, Y. Bottinga, and P. Richet, Geochim. Cosmo-
chim. Acta 46, 1061 (1982).
14. G. Hetherington, K. H. Jack, and J. C. Kennedy, Phys.
Chem. Glasses 5, 130 (1964).
15. N. M. Pavlushkin, Fundamentals of Sitalle Technology
(Stroœizdat, Moscow, 1979) [in Russian].
16. H. Scher and R. Zallen, J. Chem. Phys. 53, 3759 (1970).
17. M. I. Ozhovan, Zh. Éksp. Teor. Fiz. 104, 4021 (1993)
[JETP 77, 939 (1993)].
18. R. Bruning, J. Non-Cryst. Solids 330, 13 (2003).
Translated by E. Golyamina