POLYMERIZATION PROCESS RESEARCH
IN OXIDE MELTS BY MOLECULAR DYNAMICS
AND STATISTICAL-GEOMETRICAL METHODS
Voronova L.I., Voronov V.I., Gluboky J.V.
The method of forecasting of structure and physical-chemical properties of oxide
melts on a base of the complex simulation by a quantum-chemical method MNDO,
molecular-dynamic and statistical-geometrical methods realized by the authors by the way
of a program complex is described. The methods of simulation of structure with different
levels of an approximation are considered: the short-range order, nanostructure,
microinhomogeneity. The method of сovalent bonds network covering developed by the
authors for exploration of oxide melts polymerization processes is explicitly described.
Some results of structural simulation of the systems SiO2-CaO and SiO2-Na2O are shown.
INTRODUCTION
The oxide melts are disordered strong-interacting polymerized systems. One of the
basic problems of physical chemistry of oxide melts is the exploration of correlation of
structure and physical-chemical properties of these objects.
The absence of the sequential analytical theory of slag melts and heavy complexities
of their experimental study stimulated active development of a new scientific direction
"computer science of materials". The results of computer experiments are used for design
of materials with required properties.
Among computer methods used for study of strong-interacting systems of many
particles, methods of computer simulation have widely spread, such as the quantumchemical semiempirical methods, in particular, the method MNDO (modified neglect by
diatomic differential overlapping); Monte Carlo method (MC); a method of molecular
dynamics (MD). These methods, practically from "of the first principles", allow to obtain
various physical-chemical properties. The adequacy of obtained results is defined by
accuracy of used mathematical models.
One of actual problems, successfully explored within the framework of "computer
science of materials", is the study of the polymerization processes of multicomponent oxide
melts. It is possible to use both the MC and the MD-methods for these purposes, however
the MD-method is more preferable, because with its help it is possible not to only define
parameters of structure formation, but to explore multilaterally a fundamental problem of
“structure-property” correlations.
It is possible by results of molecular-dynamic simulation, with engaging statisticalgeometrical (SG) methods to investigate system structure with different levels of detailing
(short-range order, nanostructure, microinhomogeneity), to define features of short-range
and extended structure, to pick out characteristic building blocks and regularities of their
relative location.
452
COMPLEX MNDO-MD-SG SIMULATION
For forecasting oxide melts structure and their physical-chemical properties we use a
complex method that includes a semiempirical quantum-chemical MNDO (modified
neglect by diatomic differential overlapping) method, molecular dynamics (MD) method
and statistical-geometrical (SG) methods.
The complex method simulation phases
 MNDO simulation in a cluster approximation, obtaining the body of basic data;
 Superimposed potentials parametrization on the basic data processed by multivariate
optimization methods;
 Molecular dynamics experiment, arrays of physical chemical properties as result;
 Structure investigation by the method of сovalent bonds network covering
 Structure investigation by statistical-geometrical methods;
 Investigation of “composition-structure-property” correlation;
The simulation outcomes
 Parameters of potential functions in an ionic and an ionic-covalent approximation;
 Energy parameters;
 Thermodynamic parameters: molar heat capacities, adiabatic and isothermal
compressibilities, temperature-expansion factor;
 Transport properties: autocorrelation functions, root-mean-square displacements, diffusion
factors, volume and shear viscosity, electric conductivity, thermal conductivity;
 Spectrum characteristics: a density function of oscillatory states, infrared absorption spectrums;
 The averaged structural parameters of the short-range order: partial functions of radial and
angle distribution, bond lengths, average coordination numbers and their allocations;
 Estimation of the system polymerization degree: the distribution functions of complex
anions on some parameters, lifetime of polyanions, portion of Oxygenium of a different
type, portion of plane rings;
 Statistical- geometrical parameters of extended and long-range structure: allocation of
number of sides, volumes and squares of polyhedrons, squares of sides, spherical
tetrahedral and octahedral factors;
 Correlation dependencies of physical-chemical properties and basis nanostructure of
melts as tables and diagrams;
 Visualization of the system evolution
Ionic covalent model (ICM)
For forecasting properties of the multicomponent oxide melts containing networkforming Si, B, Al ions and improving results adequacy we use perspective ion-covalent
model (ICM) applied for systems, containing stable long-living clusters with a high share
of covalent connections [1]. The pair spherical-symmetrical long-range Coulomb
interaction, the two- and three-particle covalent interactions connected to quantum
mechanical dipole and quadrupole moment are taken into account in ICM.
The particles of modeling system – cations network-formers(CNF), cations-modifiers
(CM) and anions of oxygen(O) have
the following attributes
in ICM: mi -mass, qi–effective


charge, i – effective radius, ri - radius-vector, v i - velocity; besides there are set:
453
d0 – CNF-O bond length, 0- equilibrium O-CNF-O bond angle and force constants:
kit – two-particle covalent interaction, kit – three-particle covalent interaction.
The particle’s potential is defined by particle’s attributes and its implementation of
elementary structural unit “implication condition“. Since all particles have a charge as
attribute, The Coulomb interaction is described by pair ionic sphere-symmetric potential

 
 cor  k u l ri j =  kul r
i j
 +  cor ri j  ,
(1)
that have terms in the Pauling's form, most often used at MD-simulation of oxide melts [2].

co r  k u l
r i j  

1  1 sign q q
i j
 n 1

q i q je 2

 0 r i2j
4
 i   j

 r ij






n

,


(2)
where rij is a particle distance, e is the electron charge, n is repulsive curve steepness parameter.
The implication condition: In case the particle belongs to a "regular" elementary
complex that includes atoms of oxygen in number of atom network-creator valency,
potential (1) is supplemented by two- and three-partial covalent contributions, associated
with the quantum-mechanical dipole and quadrupole moments:
two-particle potential:  cov r m k    cov r m k  d 0    cov 1,5d 0  ,
(3)
three-partial potential:  cov  k mk    cov  k mk   0    cov 1,5 0  ,
(4)
where m is the index of silicon atom in center of an elementary structural unit; k and k' are
elementary structural unit oxygen atom indexes; qkmk is O-CNF-O equilibrium angle; 1.5d0, 1.50
are maximum covalent two- and three-particle forces action radius and angle respectively.
The introduction of the last terms in the equations (3) and (4) allows to explicitly
describe an advantage in energy under considering the covalent interactions, saving
continuity of potential function.
Covalent contributions are described in the Keating's approximation [3]:


 cov r mk  d 0 
 cov  kmk   0  
3
16 d 02
3
8d 02

2
 k i t r mk
 d 02

2
,

 
 kit rmk rmk  d 02 cos  0
(5)

2
.
(6)
Potential energy m of covalent interaction of belonging to a m-elementary complex particles:


 m    cov r m k    cov  k m k  .
k 
k  k

 


(7)
Complete potential energy of a simulated system, considering the covalent interactions
inside an elementary complex:
U    corkul r i j     m .
(8)
i j
m
Basic MNDO accounts and parametrization of potential functions
The parameters of superimposed potentials (2-6) are determined with the help of
semiempirical quantum-chemical method MNDO, based on the electronic structure
computation, which gives the most stabilized and close to the experiment results.
454
The basis of all semiempirical quantum-chemical methods is the cluster approximation.
Representative cluster or complex (fig. 1) retaining main characteristics of the system is cut
out and its Schrodinger equation is solved. The series of used approximations and
limitations are described in the special literature [4,5].
Figure 1. The representative clusters for MNDO-simulation of a system SiO2-CaO.
Some characteristics related to multiparticle interaction are defined by the MNDO-method.
Namely they are: total energies of complexes, heats of formation, energies of atomization,
effective atom charges, orbital population, equilibrium bond lengths valence angles, force
constants. However, it is impossible to use directly these basic MNDO-data in MDsimulation for some reasons. We have developed special technique that allows to interpret
the results of basic MNDO-computations in the ionic-covalent model terms [1].The
parametrization of a superimposed potential functions in ICM is made with the help of the
program of multivariate optimization. Here the force constants of two- and three-particle
covalent interaction are varied parameters, that are searched by minimization of the
criterion function. This function is built as the sum of squared forces deflections calculated
through potential gradient (2-6) and corresponding forces in a linear approximation
obtained in MNDO-calculations.
MD simulation
The “particles model” is used for molecular dynamics simulation, i.e. there is the mutual
unique dependence between physical particles and computer model particles. The classical
differential equations of motion based on Newtonian laws are solved for each particle:

d ri ( t )
dt

 vi (t)


dvi (t )
Fi (t )  mi
,
dt
(9)
which we approximate by finite-difference equations on Beeman algorithm [7]:






ri t  t   ri ( t )  v i ( t )  t  4a i t   a i ( t  t ) t 2 6 ,





v i t  t   v i (t)  t 6  2a i (t  t)  5a i (t)  a i (t  t)
(10)

Where mi, a i - mass and acceleration of particle i, Fi - complete force affected at particle i,
t - simulation time step (is usual~10-15 c). The periodic boundary conditions are used [8],
the real volume of cube depends on the melt density.
455
STRUCTURE RESEARCH TECHNIQUES
One of obtained by the MD computer experiment results is collection of coordinates
and velocities of the particles of simulated system for each configuration, that allows to
investigate both structure and dynamics of a system in detail.
The averaged structural parameters of the short-range order
The averaged structural parameters of the short-range order determined traditionally
on the base of these data are described below.
The radial distribution function (RDF) g  r  of particles of type  in an spherical
layer of radius r around particles of type  :
V n  r, r  r 
,
g  r  
N
4r 2 r
(11)
where n  r, r  r  is an average number of particles of type  , located at distance from r
till r  r from particles of a type  ; V - volume of the system.
Coordination number n  is an average number of particles located inside a sphere
with the radius equal to distance up to the first minimum of radial distribution function.
N  rmin
n 
4r 2 g  (r )dr ,
(12)

V 0
where rmi n - distance up to the first minimum of RDF: an average distance between
particles of type    (bond length or radius of the first coordination sphere).
The angles between particles     are defined for particles of type  that located inside of
the first coordination sphere of particle of type  . The cosine law is applied to calculate
angles on known coordinates of particles.
The angle distribution function (ADF) are calculated by the formula:
 n   ,   
,
(13)


g     k
 N  
k
where n   ,    is number of angles between particles      in limits from  to
   for particles of type  , located in the first coordination sphere of a particle of
type  , at step k, N   is number of all    angles at step k.
The structural parameters of the short-range order for the melt with the mole fraction
0.3 NSiO2 of binary system SiO2-CaO are indicated in fig.2 and fig.3. The square of the first
peak of the curve 1 in fig. 2a defines the coordination number of silicium on oxygen, r0 – an
average distance between atoms of two types (bond length).
However the listed averaged short-range order parameters do not contain information
about structure features, because they reflect the averaged pattern on the whole volume of a
sample. They say nothing about extended and long-range structure, which is defining in
forming of physical-chemical characteristics of slag melts, that’s why the other approaches
for exploration of structure are actively developed.
456
The computation of distributions of coordination numbers for atoms – networkcreator is informative enough [10]. It gives an opportunity to obtain parameters defining
features of the short-range order having a good agreement with the experimental data.
The basis of this technique, as well as in a classical case, is the radial distribution
functions of atoms. The tables, combining information about relative numbers of ions of
different types of this or that coordination; share of contents of elementary structural units
as function of molar ratios of slag components; contents of non-bridging oxygen as
function of glass composition are used as estimated parameters.
g () 109.1
2.5  
2
1
1.5
137.9
1
0.5
, deg
2
0
70
90
110
130
150
170
a)
b)
Figure 2. a) The radial distribution function (RDF) for 1)Si-O, 2)O-O, 3)Ca-O,
b) The angle distribution function (ADF) 1)O-Si-O, 2)Si-O-Si
o
4
L, A
3
2
1
0
2050
10
1
3
4
2
3
1
2
4
5
T, K
2200
n
0
2050
2350
T, K
2200
2350
a)
b)
Figure 3. Temperature dependences of a) average distances (bond length), b) coordination
numbers, for 1)Si-Si, 2)Si-O, 3)O-O, 4)Ca-O.
The method of Voronoy’s polyhedrons and Delone’s simplexes
For the structure detailing it is expedient to use statistical-geometrical methods, in
which with help of a special fragmentation of a system on polyhedrons it is possible to get
both regularities and features of short-range and extended structure of the explored system.
There are number of fragmentation algorithms, some of them are Tanemura's algorithm, the
method of circumscribed spheres, facets bypass method.
The most potent and deeply developed is the method of Voronoy's polyhedrons and
Delone's simplexes [11]. The fragmentation is the set of the atom centers of a system {A}.
For each center {A} it is always possible to indicate the range, which points are closer to
the given center, than to any other center of a system. These are Voronoy's polyhedrons
(fig. 4 a) the fragmentation of a system on tetrahedrons (Delone's simplexes) is
simultaneously created, the apexes of tetrahedrons are the atoms, and the sphere
circumscribed around of a tetrahedron, does not include other atoms of a system (fig. 4b).
457
A fragmentation of a system on Voronoy's polyhedrons and the Delone's simplexes are
unambiguous and fill in the whole space without hollows.
a)
b)
Figure 4. Example of construction a) of Vononoy’s area around the arbitrary centre of a
system, b) Delone’s simplex (1-2-3) -the circle, circumscribed around of atoms, does not
include other atoms
There is an unambiguous correlation between a Voronoy’s fragmentation and
Delone’s fragmentation since each apex of a Voronoy’s polyhedron is the center of
circumscribed around of a particular Delone’s simplex sphere. The explorations of structure
through Voronoy’s and Delone’s fragmentations are based on analysis of metrical and
topological performances of obtained formations. For Voronoy’s polyhedrons the following
characteristics are used: <f> - an average number of facets on a Voronoy’s polyhedron; the
V2
sphericity coefficient sph  36 3 , where S- the area of the surface of a polyhedron, V- its
S
volume; a topological index n3n4n5n6n7 …, where ni- number of i-cornered facets at this
polyhedron; the matrix of neighbourhoods {nij}, where each element nij is number
corresponding amount of j-cornered facets around of all i-cornered facets on this
polyhedron; an index of neighbourhoods 33445566 ... - diagonal elements of
neighbourhoods matrix.
As against Voronoy’s polyhedrons, the Delone’s simplexes have the same topological
type, therefore they can be distinguished among themselves only metric – by size and form.
In these purposes such characteristics are used , as tetrahedricity
T   ( l i  l j ) 2 / 15l 02 ,
i j
where lj is edge lengths of the given simplex, and l0 – an average length of its edges;
and octahedricity
O

i j
i , j m
( li  l j ) 2 / 10l 02 
 ( li  l m /
im
2 ) 2 / 5l 02 ,
where lm is the length of the longest edge of a simplex. The less values T and O, the closer
simplex to a regular tetrahedron and octahedron accordingly. The method allows to get
information about current structure of a system, various structural characteristics, as system
polymerization degree, free ions presence, passability of a system for particles of a
particular type, diffusive characteristics.
458
The covalent bonds network covering method
We have developed a “covalent bonds network covering” method for exploration of
structure of polymerizing systems [9].
The method is that for all simulated structures the complex (i.e. complex anions)
distribution functions (CDF) are obtained on various characteristic parameters рarm of
complexes T, existing in a system; рarm = Tyрe, N,  cat ,  O ,  O  ,  O  , where Tyрe - the
type of a complex, is defined by number of varied particles which are included in a
complex T (Tyрe), N - total number of particles in a complex T (N),  cat - sum of cations
network-creators in a complex T(  cat ),  O - sum of free oxygen in a complex T(  O ),
 O  - sum of non-bridging oxygen in a complex T ( O  ), O0 - sum of bridging oxygen in
a complex Т (  O0 ). It is possible to define degree of polymerization of a system on CDF of
the following kinds:
1. f T ( Tip ) 
 K T ( Tip )
i
i
 KT
j
, where f T ( parm ) is the portion of T(рarm) complexes; K T ( parm )i is the
j
number of configurations, on which the complex T ( parm )i exists, the sum in the nominator
is taken over all the complexes of this type; K T j is the number of configurations on which
any complex Tj exists, the sum in the denominator is taken over all j existing complexes.
  T ( parm )
 T ( parm ) 
i
i
, where  is
i
average lifetime of a complex T(рarm),  T ( parm)i is lifetime of a complex T(рarm)i, the
sum is taken over complexes with the same value of the parameter.
2. The lifetimes for various complexes are defined:
3. The portions of different-types oxygen atoms are calculated:
K
K
 NO 0i
D
O0
 i 1
NO  K
,
D
O

 NOi
i 1
NO  K
D
O 2
1 ( D
O0
D
O
),
where DO0 is the portion of bridging oxygen, DO  is the portion of non-bridging oxygen
and DO 2  is the portion of free oxygen in a system, N , N O  are quantities of bridging
Oi0
i
and non-bridging oxygen atoms respectively, NO is the amount of oxygen atoms in a
system, K is a total amount of system configurations in a phase.
4. The portions of closed structural units D(grouр)n of a different degree of complexity are
calculated:
D( group )n 
 N gr( n, k )
k
 N gr( n, k )
k
, where N gr(n, k) is a number of closed structural
n
units of a n-degree of complexity on k configuration.
459
Similarly it is possible to define portions of plain rings of different dimensionality:
 N ring ( n ,k )
, where N ring(n,k) - number of rings
D( ring )n  k
 N ring ( n ,k )
k
n
th
of n dimensionality on a k-configuration, and complexes charge:
Q( parm )   n   n , where n is the number of links of those particle
O  mo d i f i c
c at
types, which are indicated in inferior indexes of the sums.

DT(cat)
900
1
Si10-19
Si1
800
0.8
Si20-50
Si2
700
Si50 and more
Si3
0.6
400
Si1
Si4-9
Si2
Si10-19
Si3
Si20-50
Si4-9
Si50 and more
300
0.4
200
0.2
100
NSiO2
0
0.4
0.5
0.6
0.7
NSiO2
0
0.4
0.8
0.5
0.6
0.7
0.8
a)
b)
Figure 5. а) portions DT(cat) of complexes of a different type, b) configurational lifetime 
of complexes of the system SiO2-Na2O
Some results of investigation of the process of the polymerization in the system SiO2Na2O is shown in fig. 6. Seven melts were simulated in the whole range of compositions. At
increase of a molecular fraction SiO2 the process of large silicooxygen groupings forming,
leading to networking, is observed. Simultaneously, there is a process of their constant
regenerating, as the lifetime even of large classifications (more than 50 atoms of silicium) in
an middle region of compositions does not exceed 400 configurations. It is connected with
the fact that both the atoms of oxygen, and small-sized silicooxygen anions having 1-2 atoms
of silicium, permanently associate and abandon large complexes, owing to the migration of
atoms of sodium.
0.8
D(ring)n
1
D(group)n
2
0.6
1
0.5
0.4
0.2
2
1
4
0
0.5
0.6
0.7
0.8
3
5
NSiO2
4
3
0
0.4
0.9
0.6
0.8
5
NSiO2
a)
b)
Figure 6. Portions a) of plain rings D(ring)n with different number of cations networkformers n, b)of closed structural units D(group)n with number of cations n;
n= 1) 4, 2) 5, 3) 6, 4) 7, 5) 8.
460
In fig. 6 the portions of plain rings and portions of closed structural units of different
complexity in dependence on composition of a melt of a system SiO2-CaO are shown. The plain
rings exist only in range of compositions (0.7-0.9) NSiO2 and include from 4 up to 8 cations of
silicium. For structure 0.7 NSiO2 the pentatomic rings, and in structure 0,9 NSiO2 - fouratomic ones
dominate. The portions of rings with the major contents of cations network-creator are minor.
The closed structural units differ from rings in that the atoms, included in them, do not belong to
one plane. As follows from fig. 6, such structural formations are present already at structure 0.5
NSiO2 and have 4 or 5 silicium atoms. For a range of structures (0.7-0.9) NSiO2 their portion
decreases and larger closed units containing 6,7,8 atoms of silicium are formed.
With usage of the stated approaches we obtained series of results for oxide binary
mixtures containing atoms of Si, Al, Na, K, Ca, Mg, which were the basis for creating the
structure - property correlation dependencies. The analysis of the obtained results shows
efficiency of usage of the different approaches for structure analysis, so the obtaining of
multivariate characteristics of structural parameters allows to penetrate deeply into the
nature of polymerized oxide melts and to obtain more reliable prognoses of properties.
ACKNOWLEDGMENTS
The work is executed by support RFBR, grant № 97-03-32531
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