Investigation of free volume percolation under the liquid–glass phase
transition
Vladimir P. Voloshin, Yuri I. Naberukhin, and Nikolai N. Medvedev
Institute of Chemical Kinetics and Combustion, Novosibirsk 630090, Russia
Mu Shik Jhon
Center for Molecular Science, Korea Advanced Institute of Science and Technology, 373-1 Kusung-dong,
Yusung-gu, Taejon 305-701, Korea
~Received 11 October 1994; accepted 14 December 1994!
Mutual arrangement of large volume atomic cells ~defined as the Voronoi polyhedra! is investigated
in molecular dynamics models of Lennard-Jones liquid, crystal and amorphous solid by the
percolation theory methods on the Delaunay network. In addition to ordinary percolation some
extended formulations of the percolation problem are realized. All variants of the percolation
analysis lead to characteristics of percolating clusters and to the percolation threshold values which
do not practically differ for all the three phases. This provides evidence that the phase transition
liquid–amorphous solid is not associated with percolation through regions with a large free
volume. © 1995 American Institute of Physics.
I. INTRODUCTION
Free-volume approach is one of the most popular in describing mobility phenomena in liquids and amorphous solids ~glasses!.1–3 According to this concept, the liquid-to-glass
transition occurs when the total free volume of a system
decreases below some critical value. Developing this idea
Cohen and Grest2 divided all particles according to their free
volume into two classes—liquidlike and solidlike—and supposed that the liquid–glass transition is of a percolation character: macroscopic mobility arises as a result of the appearance of an infinite cluster of liquidlike particles, which
pervades the whole system.
Although this idea seems to be intuitively reasonable, it
needs a serious verification which can be realized by computer modeling methods. Hiwatari4 was the first to undertake
such an attempt. Using molecular dynamics models of liquids and amorphous solids he concluded that the liquid-glass
transition is indeed described by a type of percolation transition. However, this conclusion does not seem to be convincing: it was obtained in an indirect way since the author
did not perform the percolation analysis in the true sense of
the term. He did not study the properties of a percolating
cluster, neither did he calculate the percolation threshold for
liquidlike particles and in order to distinguish them he used
additional, nongeometric criteria.
On the other hand, some recent papers cast doubt on the
very concept of Cohen and Grest. Thus, studying hydrocarbon models the authors5 did not find substantial differences
in the free volume distributions in liquid and amorphous
states that should have been exist there according to this
concept. Moreover, no correlation has been observed between local mobility of a particle and its free volume in
two-dimensional models of liquid and amorphous argon.6
In this paper we undertake more detailed investigation of
the percolation through the atomic cells with the large free
volume in the models of liquid, glass, and crystal of the same
density. To this aim, the methods of the Voronoi–Delaunay
theory of the space division were used which enable us to
reduce our problem to the bond and site percolation problems on the Delaunay networks of models studied. In Sec. II
we define molecular dynamics models which we used. Two
definitions of the free volume and correlations between them
are discussed in Sec. III. The results of the percolation analysis are presented in Sec. IV where we consider both the
classic form of the percolation problem and two extended
formulations. No modification of the percolation analysis reveals any differences in percolation properties in the three
phases. Hence, we conclude in Sec. V that the liquid-glass
transition is not associated with percolation through the regions with large free volume.
II. MODELS
All our models consisted of 500 atoms interacting
through the Lennard-Jones potential w (r)5 e [(d/r) 12
22(d/r) 6 # and were generated by the molecular dynamics
procedure in a cubic box with periodic boundary conditions.
The interaction cutoff distance was at half of the box length;
the location of the potential minimum, d, was chosen as a
unit of length. The results for each phase state were averaged
over 75 models separated by 20 time steps ~Dt52 fs!.
Models of an amorphous state were obtained from the
supercooled liquid model at reduced temperature
T * 5kT/ e 50.4 and reduced density r * 5 r d 3 5 rs 3 2 21/2
50.818 by relaxation over 10 000 steps in the NPT
ensemble7 at constant temperature T * 50.2 and constant
pressure P50. The stable equilibrium density value, r*, was
achieved after about 2000 steps and was equal to 1.00 ~the
structure of these models being investigated in detail in Ref.
8!. It is sufficient to use an NPT ensemble for preparing the
amorphous state, since the temperature decrease in the NVT
ensemble brings about a pressure lowering up to negative
values, which inevitably leads to crystallization and the appearance of cracks or great holes.
In order to investigate the influence of structural differences between the phases on the free volume arrangement
and to exclude the trivial scaling factor we prepared the mod-
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Chem. Phys. 102 (12), 22 March 1995
0021-9606/95/102(12)/4981/6/$6.00
© 1995 American Institute of Physics
4981
Voloshin et al.: Liquid–glass phase transition
4982
FIG. 1. Pair correlation functions, g(r), for a crystal ~top!, a liquid ~middle!
and an amorphous solid ~bottom!.
els of a crystal and a liquid of the same density r*51.0 as
for the amorphous solid model. This was done by the molecular dynamics simulation in the NVT ensemble. The crystal model was generated from the perfect fcc structure by
relaxation at T * 51.0 over 500 time steps. To obtain a liquid,
the crystal was melted at T * 55 and then relaxed at T * 51
over 1500 steps.
Correspondence of the models to the phases discussed is
verified by the characteristic form of the pair correlation
functions ~Fig. 1, note the two-peaked second maximum
typical for the amorphous state! and by substantial differences in the atomic mobilities. The self-diffusion coefficient
estimated from the behavior of squared atomic displacements
is about two order lower in the amorphous phase than in the
liquid.
III. FREE VOLUME
There are several approaches to the definition of the free
volume. The first which was used by Hivatari4 is borrowed
from the cell model of liquid9,10
v if 5
E
V
exp~ 2F i ~ r! /kT ! dr,
~1!
where
F i ~ r! 5
1
2
(
@ w ~ u r1ri0 2r j u ! 2 w ~ u ri0 2r j u !# .
~2!
jÞ1
Function fi ~r! determines the cell potential of particle i relative to the position of its local minimum ri0 . A set of local
minima coordinates for each model defines its proper or hidden structure.11,12 Here, as in Ref. 12, we used the gradient
descent method for finding such minima. After that the volume integral in ~1! was calculated by summation over spherical layers of thickness Dr centered at ri0 , each layer being
FIG. 2. Correlation between the free volume calculated by formula ~1! and
the Voronoi polyhedron volume.
divided into segments of the same size Dr. The calculation
was over when the contribution of a subsequent layer was
1000 times lower than that of the central sphere of radius Dr.
The Dr value was chosen in such a way that its reduction by
one-half did not change the average free volume value more
than by 0.01%.
The second approach which was used by Cohen and
Grest2,3 assumes the volume of the Voronoi polyhedron of a
particle ~with the exception of its proper volume! as a measure of the free volume. This approach has many points in its
favor. It determines the cell or cage of a given particle as a
space region all the points of which are the nearest to this
particle. Such a purely geometrical definition enables one to
apply the exact Voronoi–Delaunay methods for the percolation analysis and to investigate the topology of clusters of the
Voronoi polyhedra on the Delaunay network—in much the
same way as the percolation analysis was performed earlier
on the Voronoi network for investigation of the structure.13
Furthermore, calculation of the Voronoi polyhedra needs
much less computer time ~by a factor of 100 or more! than
that of the volume integral in ~1!. Therefore we will deal here
with the Voronoi polyhedra. However, we attempted a comparison of both approaches in order to demonstrate that the
main conclusions of our work do not depend on the free
volume definition.
In Fig. 2 each atom of the model is displayed by the
point with coordinates corresponding to the Voronoi polyhedron volume of this atom along the abscissa and to the logarithm of its free volume, according to formula ~1!, along the
ordinate. We see high correlation of values of both volumes,
which is reflected in high magnitude of the correlation coefficients ~Table I!.
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Voloshin et al.: Liquid–glass phase transition
TABLE I. Comparison of the atomic free volume, V f , calculated by formula
~1! with the volume of their Voronoi polyhedra. r(x,y)5 ( i (x i 2 ^ x & )
•(y i 2 ^ y & )/( s x • s y ! is the correlation coefficient of characteristics x and y;
^ x & , ^ y & and s x , s y are their average values and variances.
Models
fcc
Crystal
Liquid
Amorphous
solid
^ V f &•104
^ln~V f !&
r(V f , V VP!
r~ln~V f ), V VP!
54.0622.8
45.3626.4
25.3160.41
25.5460.52
0.9342
0.8697
0.9565
0.9160
5.9662.61
27.5060.38
0.8720
0.9094
A significant difference between two definitions of the
free volume is in that the mean volume of the Voronoi polyhedron is unequivocally determined by the density ~and is
equal to 0.707 in each of our models!, whereas the volume
calculated by formula ~1! depends on the temperature ~see
Table I!. For the aims of the percolation analysis this temperature dependence is, however, insignificant since the laws
of mutual arrangements of particles are governed by the relations between the values of the particle free volumes and
do not depend on their concrete numerical values. Therefore,
due to the strong correlation of the free volume values according to two determinations, both of them can be used for
the percolation analysis.
IV. PERCOLATION ANALYSIS
In a disordered system all the Voronoi polyhedra ~VP!
are not identical to each other. Their variety is reflected in the
VP volume distributions ~Fig. 3! which have practically the
same shape for all the three phase states and differ only in
their variances. It is difficult to draw some conclusions on
the structural differences of the models from these distributions. Much more information can be obtained if we work
with total tessellation of the Voronoi polyhedra. To describe
the mutual arrangement of the Voronoi polyhedra in the tessellation we used here the methods of the Voronoi–Delaunay
theory of the space division into polyhedra which was repeatedly outlined in our recent papers ~see, e.g., Ref. 14!. It
should be reminded that atoms whose Voronoi polyhedra
FIG. 3. Distributions of the Voronoi polyhedron volumes in a liquid ~the
thick line, variance s52.137•1023!, a fcc crystal ~the line of medium thickness, s51.204•1023! and in an amorphous state ~the thin line,
s50.959.1023!. The average value for all distributions is the same ~^V &
50.708! since all the models are of equal density.
4983
TABLE II. Percolation characteristics of the Delaunay network for site coloring according to the Voronoi polyhedron volume.
Models
fcc
Crystal
Liquid
Amorphous
solid
Pc for random
coloring
Pc
Vc
P`
0.16960.023
0.17460.022
0.11660.038 0.75160.009 0.82960.449
0.12260.026 0.76360.010 0.79860.317
0.17660.024
0.12860.023 0.74460.005 0.80660.300
share the face are called geometrical neighbors. Connecting
the center of each atom with the centers of all its geometrical
neighbors by bonds, we obtain a network spanning the whole
space of the model which is called the Delaunay network.
Evidently the metric and topology of the Delaunay network
are unambiguously determined by the given atomic configuration in the model. Since in a disordered system the Voronoi
polyhedra have different number of faces, the different number of bonds converge at each site of the Delaunay network.
In our models about 14 bonds meet at the site on the average.
Having calculated the Voronoi polyhedra according to
the algorithm14 for all the atoms we then construct the Delaunay network for each model which is a starting point for
our analysis. Existence of the network enables one to reduce
the problem of the mutual arrangement of a particular kind
of particles, e.g., atoms with large free volumes, to the standard percolation problem of sites or bonds.
A. Coloring the Delaunay network sites according to
the Voronoi polyhedron volume
We begin percolation analysis with coloring the Delaunay network. For our aim we select ~color! those network
sites for which the volume of the corresponding Voronoi
polyhedron exceeds some preassigned boundary value V b .
Neighboring on the network ~contiguous! colored sites combine into clusters whose properties are the subject of percolation theory. Interesting for us are here first of all the percolation threshold Pc @i.e., minimum concentration of the
colored sites ~atoms! at which a percolation cluster appears
extended across the entire model# and corresponding to it
boundary value of the Voronoi polyhedron volume—the
critical volume V c .
The Pc and V c values are presented for each model in
Table II. We see that the critical VP volumes at the percolation threshold, V c , differ in the three phases not significantly
but beyond the errors. Thus, we can indicate such a boundary
value of the volume ~par example, V b 50.755! at which a
percolation cluster of atoms with the Voronoi polyhedron
volumes greater than this boundary will be observed in the
model of a liquid whereas in a crystal and in an amorphous
solid only finite clusters exist. Hence, it seems to be possible
to distinguish, in accordance with Cohen and Grest, liquidlike atoms ~with V.V b ! from solidlike ones ~with V,V b !.
However, such a definition would be formal since the percolation thresholds Pc do not differ within the accuracy in all
the three phase states ~see Table II!. This means that the
tendency of atoms with the largest VP volumes to join into a
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Voloshin et al.: Liquid–glass phase transition
percolation cluster in a liquid is not greater than in crystals or
glasses.
The revealed coincidence of percolation thresholds in
the three phases is surprising the more so, as they differ
essentially from the percolation thresholds at the random coloring of the same networks ~Table II!. The coincidence of
Pc ’s as well as of other percolation characteristics, such as
the fraction of sites belonging to the percolation cluster, P` ,
~Table II! cannot be considered accidental; it reflects some
structural correlations characteristic of all the three condensed phases.
Coincidence of the percolation thresholds also means
that differences in the values of the critical volume V c in the
three phases at the same packing density result exclusively
from the differences in the variances of the VP volume distributions. However, one can easily generate by computer the
quasiequilibrium models of a crystal and a liquid with equal
variances of the VP volume distributions ~e.g., an overheated
crystal and a supercooled liquid!. Therefore we have no
grounds to choose a reasonable value for V c which would be
based on intrinsic differences in the structure of the three
phases. Hence, we must conclude that at least the phase transition liquid–crystal has no relation to the percolation of liquidlike molecules.
B. Joint site and bond coloring of the Delaunay
network
The coloring procedure described above specifies the
simplest, formal setting up of the percolation problem. We
can now complicate it to study more ‘‘physical’’ relationships in the arrangements of atoms with large VP volumes.
First of all it should be noted that the bonds of the Delaunay
network are not equivalent. Among the geometrical neighbors of a given atom there are both closer ones which give
rise to larger faces on the Voronoi polyhedron of this atom
and more distant ones corresponding to smaller faces which
appear for a short period of time and cannot be assigned to
real contacts between atoms. Therefore it is possible that in
an amorphous solid or crystal the long bonds between distant
neighbors join individual distant liquidlike clusters into a
unified one only formally. This masking effect of long bonds
could make the percolation threshold in the models of solids
equal to that in the liquid. To remove this effect of long
bonds we suggest joint coloring of the Delaunay network:
simultaneous coloring of both the sites—according to the
corresponding Voronoi polyhedron volumes—and the
bonds—according to their length. In other words, this can be
formulated as the site problem on the reduced Delaunay network in which the longest bonds ~with l.l b ! are removed.
Since the boundary value of the bond length l b cannot be
chosen unambiguously we performed such a joint coloring
by varying these values over a wide range.
Figure 4 presents the dependence of the VP volume at
the percolation threshold, V c , on the boundary length of
bonds. Decrease of this length leads to a noticeable decrease
in V c for all models. However, V c decreases most quickly in
the model of a liquid; this means that in a liquid bonds between distant neighbors are involved in the percolation clusters to the same extent as in solid phases. This conclusion is
FIG. 4. Dependence of the Voronoi polyhedron volume at the percolation
threshold, V c , on the boundary value of the colored bond length, l b , for
joint coloring of the Delaunay network. Here and in the following figures
squares correspond to a liquid, triangles—to a fcc crystal, and circles—to an
amorphous state.
also supported by Fig. 5 where we see that the percolation
threshold remains the same for all the models on a wide
interval of the boundary length. Only for the fcc crystal
model the percolation threshold differs from other models
when l b ,1.2.
C. Z -coloring of the Delaunay network sites
Cohen and Grest2,3 supposed that molecular mobility is
connected with percolation of only those liquidlike particles
which have a sufficiently large number ~z and more! of other
liquidlike particles as neighbors. This defines a new type of a
percolation problem which we will call z-percolation problem. In z problem, percolation is considered to be taking
place in the system only if there exists an ‘‘infinite’’ ~percolating! cluster of atoms each of which has at least z neighbors with the free volume larger than a critical value. Realized by a computer, z problem reduces to the double coloring
of sites of the Delaunay network: at first we color all the sites
whose VP volume exceeds some prescribed value ~concentration of such sites is p!, secondly we color once again those
of them which have no less than z colored neighbors ~their
concentration will be p z !. From this point of view an ordi-
FIG. 5. Dependence of the percolation threshold, Pc , on the boundary
length for the ordinary percolation problem ~z50). Here and in the following figures full marks correspond to correlated coloring and open marks—to
random coloring.
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Voloshin et al.: Liquid–glass phase transition
FIG. 6. Dependence of the critical volume of the Voronoi polyhedron on z
in z percolation problem.
nary percolation problem corresponds to z50, when p5 p z .
Since percolation can proceed only through the sites for
which z is no less than two, the percolation threshold p cz in
a z problem will be equal to the percolation threshold in a
usual percolation problem when z50, 1 or 2.
Results of z analysis are presented in Figs. 6 – 8. Behavior of the critical volume ~Fig. 6! and the percolation threshold ~Fig. 7! shows that the difference between the values for
an amorphous solid, on the one hand, and a liquid and a
crystal, on the other hand, increases with the rise in z. Thus,
a z dependence cannot distinguish a liquid among two solid
phases. We see also in Fig. 7 that the curve for an amorphous
solid shifts to the curves for random coloring. This effect is
seen more clearly in Fig. 8 where the behavior of the concentrations of z colored atoms, p z , is displayed at the percolation threshold, i.e., the critical concentrations of precisely
those particles which have no less than z neighbors with the
VP volume above the critical value V c . We see that the curve
for an amorphous solid breaks away from other correlated
colorings and merges with random colorings as z increases.
Such a behavior undoubtedly reflects some property that distinguishes liquid and amorphous states. However, it has no
relation to the atomic mobility since the nonflowing crystal
curve in this figure approaches the liquid curve and not the
amorphous phase curve.
Finally, we tried to use joint coloring of sites and bonds
in z-percolation problem as we did it in Sec. IV B for a
FIG. 7. Behavior of the percolation threshold in z percolation problem.
4985
FIG. 8. Behavior of the critical concentration of z colored atoms at the
percolation threshold. Note that p c is the concentration of all the atoms with
the VP volume above the critical value V c , whereas p cz is the concentration
of those of them which have at least z neighbors with V.V c .
standard percolation. But that gave nothing new. Taking
shorter and shorter bonds ~at z57! we obtained an increase
in differences between the liquid and the fcc crystal but with
a simultaneous decrease in differences between a liquid and
an amorphous solid ~Fig. 9!.
V. CONCLUSION
Hence, no formulation of the percolation problems
~some of which are rather sophisticated! allows one to reveal
such characteristics of the percolation cluster that would distinguish a liquid from both an amorphous solid and a crystal.
We believe this demonstrates clearly that percolation through
molecular cells with the large free volume is not a direct
cause of macroscopic mobility. This can be understood if we
take into account that within a great percolation cluster all
the liquidlike molecules cannot move simultaneously. For a
diffusion to take place, it is sufficient that from time to time
each molecule should take part in a collective motion of a
relatively small number of particles, the molecule groupings
being constantly changed.
We think that our results by no means discredit the very
concept of a free volume. A free volume is, of course, necessary for the existence of macroscopic mobility. We argue
only that fluidity is not related with the appearance of perco-
FIG. 9. Behavior of the critical concentration of z colored atoms at the
percolation threshold in z percolation problem combined with joint site and
bond coloring ~for z57).
J. Chem. Phys., Vol. 102, No. 12, 22 March 1995
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4986
Voloshin et al.: Liquid–glass phase transition
lation through the regions with the large free volume. At the
same time it does not mean that phase transitions cannot be
accompanied by some percolation phenomena. On the contrary, we found earlier that structural differences between a
liquid and an amorphous solid can be described in terms of
percolation.15–17 The distinguishing feature of an amorphous
substance is the existence of a percolation cluster of atomic
configurations in the form of slightly distorted tetrahedra,
i.e., the closest packing of atoms. In contrast, in a liquid
there exist percolating clusters of interstitial cavities which
are not observed in a solid phase.
ACKNOWLEDGMENTS
This work was supported in part by the Russian Basic
Research Foundation, Grant No. 93-03-5011 and Korea Science and Engineering Foundation. The authors are grateful to
Mrs. Irene Kotliarevski for her kind assistance with the English translation.
1
2
R. Zallen, The Physics of Amorphous Solids ~Wiley, New York, 1983!.
M. H. Cohen and G. S. Grest, Phys. Rev. B 20, 1077 ~1979!.
G. S. Grest and M. H. Cohen, Adv. Chem. Phys. 48, 455 ~1981!.
Y. Hiwatari, J. Chem. Phys. 76, 5502 ~1982!.
5
D. Rigby and R. J. Roe, Macromolecules 23, 5312 ~1990!.
6
M. I. Kotelyansky, M. A. Mazo, A. G. Grivtsov, and E. F. Oleynik, Phys.
Status Solidi B 166, 25 ~1991!.
7
H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and
J. R. Haak, J. Chem. Phys. 81, 3684 ~1984!.
8
N. N. Medvedev, Yu. I. Naberukhin, and V. A. Luchnikov, Zh. Strukt.
Khim. 35, No. 1, 53 ~1994! ~in Russian!.
9
J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of
Gases and Liquids ~Wiley, New York, 1954!.
10
M. S. Jhon and H. Eyring, Theoretical Chemistry: Advances and Perspectives ~Academic, New York, 1978!, Vol. 3, p. 55.
11
F. H. Stillinger and T. A. Weber, Phys. Rev. A 25, 978 ~1982!; Science
225, 983 ~1984!.
12
Yu. I. Naberukhin, V. P. Voloshin, and N. N. Medvedev, Melts 1, No. 2, 71
~1987! ~in Russian!.
13
Yu. I. Naberukhin, V. P. Voloshin, and N. N. Medvedev, Mol. Phys. 73,
917 ~1991!.
14
N. N. Medvedev, J. Comput. Phys. 67, 223 ~1986!.
15
N. N. Medvedev, A. Geiger, and W. Brostow, J. Chem. Phys. 93, 8337
~1990!.
16
N. N. Medvedev, Yu. I. Naberukhin, and V. P. Voloshin, Zh. Fiz. Khim. 66,
163 ~1992! ~in Russian!.
17
V. P. Voloshin and Yu. I. Naberukhin, Zh. Strukt. Khim. 34, No. 5, 94
~1993!.
3
4
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