Polymerization process Research in oxide melts by molecular dynamics and statistical-geometrical methods
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- Introduction
- The complex method simulation phases
- The simulation outcomes
- Basic MNDO accounts and parametrization of potential functions
- MD simulation
- Structure research techniques
- The covalent bonds network covering method
- The work is executed by support RFBR, grant № 97-03-32531
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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. IntroductionThe 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 quantum-chemical 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 statistical-geometrical (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. 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
The simulation outcomes
Ionic covalent model (ICM) For forecasting properties of the multicomponent oxide melts containing network-forming 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, 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  = + , (1)
that have terms in the Pauling's form, most often used at MD-simulation of oxide melts [2].  , (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: three-partial potential: 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, 1.50 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]:  , (5)
 . (6)
Potential energy m of covalent interaction of belonging to a m-elementary complex particles:  . (7)
Complete potential energy of a simulated system, considering the covalent interactions inside an elementary complex:  . (8)
Basic MNDO accounts and parametrization of potential functionsThe 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. 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].
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:   , (9)
which we approximate by finite-difference equations on Beeman algorithm [7]:  , 
 (10)
Where mi, Structure research techniquesOne 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)  , (11)
where Coordination number where The angles between particles The angle distribution function (ADF) are calculated by the formula:  , (13)
where 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. The computation of distributions of coordination numbers for atoms – network-creator 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. a a 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). 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:  , where S- the area of the surface of a polyhedron, V- its 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  - 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  ,
where lj is edge lengths of the given simplex, and l0 – an average length of its edges; and octahedricity
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. The covalent bonds network covering methodWe 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, 1. 2. The lifetimes for various complexes are defined: 3. The portions of different-types oxygen atoms are calculated:  ,   ,
where 4. The portions of closed structural units D(grouр)n of a different degree of complexity are calculated: units of a n-degree of complexity on k configuration. Similarly it is possible to define portions of plain rings of different dimensionality:  , where N ring(n,k) - number of rings
of nth dimensionality on a k-configuration, and complexes charge:  , where n is the number of links of those particle types, which are indicated in inferior indexes of the sums.
  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 SiO2-Na2O 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. a) b)
n= 1) 4, 2) 5, 3) 6, 4) 7, 5) 8. 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 № 97-03-32531REFERENCES
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- radius-vector, 
- velocity; besides there are set: 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.
, (3)
, (4)
- mass and acceleration of particle i, 
- complete 
- 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. 
of particles of type 
in an spherical layer of radius r around particles of type 
:
is an average number of particles of type
from particles of a type
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.
, (12)
- distance up to the first minimum of RDF: an average distance between particles of type 
(bond length or radius of the first coordination sphere).
are defined for particles of type
. The cosine law is applied to calculate angles on known coordinates of particles.
is number of angles between particles 
in limits from 
to 
for particles of type
is number of all 
angles at step k.
) b)
) b)
,
, 
, 
, 
), 
- sum of free oxygen in a complex T(
), 
- sum of bridging oxygen in a complex Т (
). It is possible to define degree of polymerization of a system on CDF of the following kinds:
, where 
is the portion of T(рarm) complexes; 
is the number of configurations, on which the complex 
exists, the sum in the nominator is taken over all the complexes of this type; 
is the number of configurations 
, where 
is average lifetime of a complex T(рarm), 
is lifetime of a complex T(рarm)i, the sum is taken over complexes with the same value of the parameter.
is the portion of bridging oxygen, 
is the portion of non-bridging oxygen and 
is the portion of free oxygen in a system, 
, 
are quantities of bridging 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.
, where N gr(n, k) is a number of closed structural
igure 6. Portions a) of plain rings D(ring)n with different number of cations network-formers n, b)of closed structural units D(group)n with number of cations n;