Molecular dynamics simulation of thermodynamic and ...

Journal of Non-Crystalline Solids 448 (2016) 16–26

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Molecular dynamics simulation of thermodynamic and structural properties of silicate glass: Effect of the alkali oxide modifiers H. Jabraoui a,⁎, E.M. Achhal a, A. Hasnaoui b, J.-L. Garden c, Y. Vaills d,⁎, S. Ouaskit a,⁎ a

Laboratoire physique de la matière condensée, Faculté des sciences Ben M'sik, Université Hassan II de Casablanca, Maroc Univ. Hassan 1, Laboratoire LS3M, Faculté Poly disciplinaire de Khouribga, 26000 Settat, Maroc Institut Néel, CNRS et Université Joseph Fourier, BP 166, 38042 Grenoble Cedex 9, France d Université d'Orléans, CEMHTI – CNRS UPR 3079, Avenue du Parc Floral, BP 6749, 45067 Orléans Cedex 2, France b c

a r t i c l e

i n f o

Article history: Received 5 April 2016 Received in revised form 21 June 2016 Accepted 23 June 2016 Available online xxxx Keywords: Silicate glass Alkali oxides Glass transition Structural properties

a b s t r a c t Molecular dynamics simulation was applied to elucidate the effect of adding alkali oxides (M2O)X(SiO2)(1 − X)with M = (Na, Li or K) into silicate glass matrix. We are interested in the study of this effect particularly on structural and thermodynamic properties of the material. Some interesting results were obtained given a new insight on the bridging process and its reliability to the observed depolymerization phenomena affecting the existing Si\\O network and depending on both the kind of the alkali modifier and its molar fraction. We observed that the thermodynamic properties are influenced by these structural modifications. Indeed, the glass transition temperature Tg has been found to decrease as the molar fraction of modifier increases depending strongly on the alkali modifier kind. On the other hand, we extracted the fictive temperature from the calculated total energy of the system and determined the glass transition by studying the variation of the fictive temperature as a function of the conventional one using different cooling and heating rates. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Silicate glasses and their properties (structural and thermodynamic) are usual topics of condensed matter physics, glass science and materials chemistry. These materials were studied by many physicists and chemists by means of both theoretical and experimental methods as a fundamental and complex problem. Time domain of many experiments through the glass transformation region routinely comes across with non-equilibrium state of the material. The fictive temperature is a common tool to characterize the non-equilibrium state (glassy state) and it is defined as the temperature at which the non-equilibrium values of a macroscopic property (e.g. enthalpy) would equal the equilibrium ones [1]. Alkali elements (e.g. K+, Na+, L+, …) are becoming increasingly essential for improving glasses materials with a greater focus on end-user application requirements, reduction of development costs, and a decrease in the time to market [2,3]. Therefore, alkali silicate glasses are the prototype of multi-component silicate glasses that find wide applications in the glass industry, photonic devices, biomaterials and microelectronics [4]. Recently, the binary alkali silicate glasses have taken much attention not only like an archetype of glassy materials, but also because of its anomalous structural, mechanical, thermal properties [3] and glass transition temperature [5,6]. For example, describing ⁎ Corresponding authors. E-mail addresses: [email protected] (H. Jabraoui), [email protected] (Y. Vaills), [email protected] (S. Ouaskit).

http://dx.doi.org/10.1016/j.jnoncrysol.2016.06.030 0022-3093/© 2016 Elsevier B.V. All rights reserved.

properties of the glassy state and predicting the glass transition temperature Tg of binary and multicomponent mixtures of molecular and ionic compounds is still poorly understood [7,8]. The present paper represents a modest attempt to provide representative information on this problem. For silica glass, the number density of Si\\O\\Si bonds is related to the degree of network polymerization; higher polymerization usually offers the specific properties that are changed by the effect of alkali modifiers. These elements act as bond breakers in the silica network, showing a decrease of the polymerization degree by transforming part of bridging oxygen atoms (BOs) into non-bridging oxygen atoms (NBOs) [9]. Glass transition temperature, being itself a useful parameter in glass technology, is sensitive to the modification in structure, which can happen due to compositional changes [10]. The aim of this present work, is the understanding of the processes involved in the alkali silicate glassy state and the study of its physical and chemical properties. In many cases these properties are not always easily accessible from experiments, so we need to develop theoretical models and/or simulations in order to investigate the phenomena related to the alkali effects. Classical Molecular Dynamics (CMD) simulation is one of the methods that is generally applied successfully in silicate glass study to shed light on the microscopic processes involved in the transition. However, the quality of a simulation strongly depends on the atom–atom interaction potential [11]. Indeed the choice of the interatomic potential type, describing the interaction between the different constituents in the simulation, is determinant for obtaining a reliable initial structure in which

H. Jabraoui et al. / Journal of Non-Crystalline Solids 448 (2016) 16–26

17

the glassy state will be prepared. Another important feature for the MD simulation of glassy systems is related to the dynamics and the rate of cooling and heating of the system and its effect on the glass transition. In this work, we emphasis on the use of MD to study alkali addition effect on silicate glass proprieties. Born-Mayer-Huggins with long-range coulombic formula [12] is used to compute pairwise interactions in the material. We use the potential parameters developed by Habasaki et al. for describing the silicate materials [13]. We focused on the effect the cooling rate and alkali oxide content on the glass transition to obtain the silicate glassy states. On the other hand, we are interested in studying the structure results of silica and alkali-silicate glasses and their relationship with the change of glass transition temperature. This paper is arranged as follows. Firstly it starts with introduction followed by a brief description of the used interatomic potential. In Section 3, we describe the used simulation and the glass making method. In the results section, we are interested in characterizing the nonequilibrium (glassy state) by fictive temperature notion in order to show the cooling rate effect on the glass transition temperature Tg. We then present the alkali oxide (Li2O, Na2O and K2O) addition effects on Tg and on the silica glass structures. The discussion of all results and conclusion of this work are given in Sections 4 and 5, respectively.

material matrix which consists of silica. The formulation of our chemical compound is written as (M2O)X(SiO2)(1−X); with M: (Na, Li or K) and X represents the molar fraction of the modifier. The X values taken in our simulation vary from 0 to 33%. All the simulations were performed with the LAMMPS package [19], using periodic boundary conditions and an integration time-step of 1 fs. Coulomb interactions were evaluated using a cutoff distance equal to 12 Å. The short-range interaction cutoff was chosen to be 8.0 Å. We know that the choice of these values is very important in molecular dynamics; we note that the values of the used cutoff distances are often omitted in publications for silicate glasses [17]. We started our simulation by placing 2400 atoms as two layers each one contains a given chemical constituent. Then we equilibrated this system for 1 ns by using NVT ensemble at temperature of 300 K. After that, the material is equilibrated at temperature between 3600 K and 5000 K using NPT ensemble for 1 ns. This temperature interval is chosen higher than the melting temperature to insure the fusion of the system. At this temperature range the system reaches a steady state where it loses its memory. Finally we obtain the glassy state by quenching our system from high temperature (liquid) to room temperature 300 K with a cooling rate of 1 K/ps. All atomic visualizations have been done using OVITO [20].

2. The used interatomic potential

4. Results and analyses

We adopt, in this work, a pair-wise additive effective potentials [14– 15] to model the interatomic interactions. The potential is based on the Born–Mayer–Huggins potential [10] which has the general analytical form:

4.1. Fictive temperature determination and glass transition temperature of silica material

qi q j þ f 0 bi þ b j exp U ij r ij ¼ 4πε0 r ij

!

ai þ a j −r ij ci c j Di D j − 6þ 8 r ij rij bi þ b j

ð1Þ

where qi is the fictive charge number for each atom (Si, O, Na, K and Li) given in Table 1. The charge neutrality of the total system is imposed to the system by the relationship between molar fractions of alkali oxide and oxygen charge in accordance to Eq. (2). rij is the interatomic distance and f0 is a normalization constant or standard force equal to f0 = 1 Kcal° [16]. a, b, c and D are parameters characterizing the material. The two terms containing r ij 6 and rij 8 represent respectively dipoledipole and dipole-quadrupole dispersion energies. The exponential repulsion term allows us to find a good agreement with experiment [14]. This potential form was used by Bauchy [17] and has shown hopeful results in the study of aluminum and calcium modified silicate properties by means of MD simulations. For our alkali silicate materials we use the parameters developed by Habasaki et al. [11] where the potential of silica glass was proposed by Matsui [18] and was adapted to alkali silicate. All parameters are presented in Table 2. The parameter D is zero for all atoms in the original version of the potential. To ensure the electro-neutrality of the whole system the different charges are linked to each other by the following relation: qO ¼

ð1−XÞqSi þ ð2 qM XÞ ð2−XÞ

ð2Þ

where X is the molar fraction of the alkali oxide in the silicate material. 3. Simulation technique As mentioned above, our system contains two constituents; alkali oxides which play the role of the modifiers (alkali oxides) and raw Table 1 Effective charges for the different atoms used in this work [16].

Effective charge

As mentioned above, we used 2400 atoms matrix of silicate SiO2 which is initially generated by placing the atoms in two adjacent surfaces, one contains silicon atoms and the other contains oxygen atoms Fig. 1a. This allowed us to obtain a 3-dimension structure of silicate crystal (Fig. 1b). The system is gradually equilibrated for 1 ns in order to reach liquid equilibrium state. The caloric curves of total energy, as a function of temperature, presents many interesting proprieties directly related to quenching (cooling) process. After this simulation experiment, the obtained liquid is quenched by a cooling rate of 1 K/ps and 0.1 K/ps. The dependence of the fictive temperature Tf (extracted from the caloric curve) as function of conventional temperature is marked by a linear dependence with a slope change at the glass transition temperature. This concept of the fictive temperature Tf can be extracted easily from the caloric curves corresponding to cooling. Tg parameter can be defined, in this case, as the glass transition temperature, which manifests itself by a change of the slope of the total energy as a function of the temperature during the cooling process [21–25]. The fitting correlation coefficient of both total energy curve limits corresponding to low and high temperatures is N 0.99. In this part of the work, we are interested in extracting the fictive temperature through the variation of total energy, in order to define the glass transition. The system is cooled to temperatures below the transition region and the fictive temperature is determined at low temperatures where the structural relaxation becomes too slow to be detected experimentally as shown in Fig. 2 [26]. During the fictive temperature variation, we can distinguish two limits as was described by Cornelius et al. [26]: - At high temperatures (liquid state) above the transition region (or transition temperature Tg),

lim T f ¼ T

TNNT g

- At low temperatures (glassy state) well below the transition region,

Si

O

Li

Na

K

2.4

X

0.87

0.88

0.85

lim T f ¼ T Lf

TbbT g

18


Table 2 Two-body coefficients for Habasaki's potential [16].

Si-Si Si-O O-O Li-Li K-K Na-Na Li-Si K-Si Na-Si Na-O Li-O K-O

A = f0(bi + bj)(eV)

ρ = (bi + bj) (Å)

σ = (ai + aj) (Å)

C = cicjeV/Å6

D = DiDjeV/Å12

0.0028251 0.00896593 0.0151067 0.00629606 0.00873353 0.00727646 0.00456058 0.00579127 0.00505078 0.01119161 0.01070141 0.01198410

0.0657 0.20851 0.35132 0.14642 0.2014 0.16922 0.10606 0.13355 0.11746 0.26027 0.24887 0.27636

1.7376 2.9162 4.0948 2.031 2.8162 2.161 1.8843 2.2769 1.9493 3.1279 3.0626 3.4555

23.1044332 69.9590392 212.9332867 5.0807467 0.0 0.0 10.8345638 0.0 0.0 0.0 32.89167 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

where TLf , is the limiting value of Tf, obtained when the glass is cooled through the transition region at a given rate. Therefore, the internal energy values increase proportionally with increasing the cooling rate values. Tg of silica glass, determined in our simulation, is found to be around 2160 ± 10 and 1940 ± 10 K for cooling rates 1.0 and 0.1 K/ps, respectively. This is in accordance with the results already found in another molecular dynamics simulation [27]. Estimates of the temperature at which simple liquids would start the formation of glassy state if either crystallization or nucleation were precluded [7]. Tg is found to be around two-thirds of Tm [28–30]. Otherwise, the glass transition temperature is in general much higher than the experimental values 1446 K [31] 1607 K [32]. This behavior has been explained by Raffaele et al. [27] and was attributed to the molecular dynamics very low time scale and to the extremely high cooling rates. In general, the glassy material during the cooling process can take three different states related to the production of the entropy as defined already by J-L Garden et al. [33]: - Well above Tg, even though silica is quenched in the case of a glassforming liquid, there is no production of the entropy because the relaxation time of the configuration degree of freedom is too small; consequently the system is considered at equilibrium and its temperature is well defined. - Around Tg, in the relaxation range, there is production of entropy and the temperature of the system is not well defined. - Below Tg, the silica system reaches glassy state, the order parameter is completely frozen, there is no production of entropy and the temperature of the glass is well determined.

At the atomic scale, our simulations show that, for silica glass, there exist BOs building tetrahedral structural units surrounding silicon atoms, as presented in Fig. 3. This structure of glassy state is similar to that found in the crystal state as has been reported by many authors [34–35]. The only difference lays in the connectivity and the compacity of these tetrahedral structural units. 4.2. Effect of alkali oxides on the glass transition temperature The various alkali modifiers play an important role of improving some structural and interacting proprieties of glassy systems. We are interested in the influence of the alkali oxide modifiers (M2O: M = K, Na and Li), which will be introduced with various molar fractions in the silicate matrix. In this part of our study, we studied the effect of these oxides on the glass transition temperature. The simulation results, presented in Fig. 4, show that each kind of alkali oxide has a specific effect on the glass transition and on the potential energy variation of silicate system. For all fitting curves in Fig. 4, to determine the glass transition temperature, a correlation coefficient of R2 = 0.99 is found. Even though we modified the matrix silicate by the same molar fractions, we observe that the smallest oxide makes the silicate system less energetic than the bigger one. Using the same molar fraction we find that Tg is also increased by increasing size of alkali oxide. Generally, Tg values of silicate systems show an agreement with other results predicted by molecular dynamics simulation [22,36]. These values are changed by alkali modifiers effect

Fig. 1. Snapshots showing the simulation sample construction. a) Two adjacent layers of Si: red atoms and O: blue atoms. b) The relaxed system at the first step with constant-NVT MD simulation at T = 300 K. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


Fig. 2. a) Variation a property “p” as function of temperature, where “p” is a macroscopic property of a vitreous system, through this variation the extracting of fictive temperature is possible. b) Variation of fictive temperature of the system as a function of the temperature during cooling of the melt. Blue and gray lines are linear fits for high temperature and low temperature fictive temperature data. The vertical lines mark the intersections between the linear fits for both cooling rates. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

toward small values as indicated in Table 3. An experimental study of tellurite glasses [37] reported that the Tg is decreased by alkali oxides effect but the effect of potassium modifier on Tg is less important than those of sodium and lithium. In fact, in the silicate glass, tetrahedral units can be connected to each other by oxygen atoms. When the

19

silicate system is modified by alkali oxide modifiers, the structure is changed and this affects the connectivity between tetrahedral units. This structural change can be related to interacting forces between three constituents (silicon, alkali, and oxygen) and consequently influenced the glass transition temperature, which is dependent on alkalisilicate system. Tg seems to increase by reducing the molecular weight [38], where the size of the basic unit is no longer well defined from the moment that the silica glass SiO2 network is strongly perturbed by the modifier. The correlation between Tg and the average mass has been established by Heuer et al. [38]. Indeed, a linear behavior in the form of Tg = Cgmυ2 (where Cgis estimated to 0.014, m is the average elementary mass and v is the velocity of sound) [39]. Also the behavior of Tg depending on ion radii has shown the same trend where Tg decreases from Li to K [40]. This change has been correlated to a decreasing rigidity of the network in the same order. This discussion was made for alkali borate glasses [41]. The glass transition temperatures obtained in the present work as shown in Table 3 or by other simulations are always higher than the experimental results [41,13,42]. The rate used at high temperatures is the slowest one allowed by our computer-time constraints. On the time scale of our simulations no rearrangement of the network structure of all compounds takes place at the experimental glass transition temperature. Thus the higher cooling rate at lower temperatures saves a considerable amount of computer time without affecting the glassy state. The change of Tg depending on the modifier constituent was reported by experimental data [10], and was attributed to a chained structure into the glassy matrix [43]. Habasak et al. [44] indicated that Tg defined by the inflection point well corresponds to that defined by the geometrical changes of coordination polyhedral. Now, we are interested to the effect of alkali oxide molar fraction on the silicate glass transition temperature Tg; we observed systematically that Tg decreases when the molar fraction of the modifier increases. Fig. 5 shows the change of Tg depending on molar fraction of (K2O,Na2O and Li2O). We notice that our results are in accordance with the experimental observations reported by many authors [41,45– 48]. The more important feature in the Fig. 5 is related to the behavior of Tg as function of Na2O molar fraction. We observe that Tg presents two distinct regimes: the first one shows linear decrease in the interval x ≤ 20%, while Tg is quasi-constant for x ≥20%. This behavior is specific to the case of Na2O modifier. This effect was already observed experimentally by Y. Vaills et al. [49], who measured elastic constants by Brillouin scattering and have shown specific effect of sodium addition in silicate glasses.

Fig. 3. Snapshots showing the structure of the silicate material at T = 300 K obtained with a cooling rate of 1 K/ps where red atoms represent silicon and blue atoms stand for oxygen. a) The box that represents the whole sample and b) a cut showing local tetrahedral units. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

20


Fig. 4. Variation of the total energy during heating and cooling of the oxide modified silicate glass, a) (K2O)33(SiO2)67, b) (Na2O)33(SiO2)67 and c) (Li2O)33(SiO2)67. For all curves the glass transition temperature is determined by the broken lines or intersection between two linear fitting curves according to both high and low temperatures. The fitting correlation coefficient of all fitting curves is N0.99.

The effect of modifier on Tg is explained by the depolymerization of the silica network by the alkali ions. This depolymerization manifests itself by the presence of long alkali chains inside the silicate matrix. This is correlated to the composition at which the silicate coordination number begins to decrease and (Li, Na or K)\\BOs bonds start to form [50–53]. Another work reported that the glass transition change induced by heavier network modifiers are more effective in reducing Tg than lighter ones [48]. Farnan el al. [35], reported that the structural relaxation is progressively inhibited before glass transition, also explained this effect. Therefore, the effect of the strength of oxygen bonds and the modification of Tg applies an influence on the silicate glass structure [36]. 4.3. Structural results It's important to explain some results presented above by correlating thermodynamic and structural proprieties. For this purpose, we used the radial pair distribution function (PDF) to characterize silicate structure obtained by molecular dynamic simulation and compared our results with experimental data (X-ray diffraction (XRD), Neutron diffraction). The radial function is very useful for understanding the global system structure and gives local insight in the vicinity of each constituent at different distances (short, medium and long range). It permits also to understand the relative values of the temperatures usually used for melting, quenching and annealing. Our aim is the understanding of the silicate glass and alkali silicate glass structures and assessing the quality of our simulation by comparing with experimental results. Fig. 6 shows the computed total PDFs of silicate system and alkali silicate systems (sodium silicate glass (Na2O)33(SiO2)67, lithium silicate glass (Li2O)33(SiO2)67 and potassium silicate glass (K2O)33(SiO2)67. In all three alkali silicate materials a 0.33 M fraction of oxide modifier is used. Fig. 6 shows that each material has special radial function either in glassy state or in liquid state. We observe also that the radial function curve of each system shows a difference between glassy and liquid states. Similarities between each state are noticed only at the first peak, whereas the other peaks change shapes and disappear in the liquid state. The change of the glass transition temperature values when alkali oxides perturb the system can be linked to the change of its structure as shown by the shape of PDF. For more insight on the structure, we extend our study to local range order through the partial pair distribution functions, as shown in Fig. 7. From these PDFs we extract the different bond distances rij and the coordination of each atom type. The results are summarized in Table 4 together with experimental results using X-ray diffraction [57] and neutron diffraction [57]. For the Si\\O pair correlation, we remark that gSi-O(r) (partial radial distribution

function) is almost the same for all the alkali-silicate glasses and is lower than that of silica glass. We also note that the first peak, related to Si\\O pair in the partial function, coincides with the first peak of the total radial function of all alkali-silicate compounds. The coordination numbers are in agreement with those reported in experiments as indicated in Table 4. In fact, in the case of silica glass, four oxygen atoms with a distance of ~ 1.62 Å surround the silicon atoms at short distance as shown above in Fig. 3b, which is in accordance with crystal structure [54]. When the silicate glasses are modified by alkali oxides, we found also that silicon forms tetrahedral units where each Si atom is surrounded by four oxygen atoms. These tetrahedral clusters are connected to each other with oxygen atoms (bridging Si\\O\\Si) but in other configurations one or more oxygen atoms are substituted by alkali-atoms M (non-bridging Si\\O\\M) (with M = Na, Li or K). On the other hand, Si\\BOs bond and Si\\NBOs bond are not equal. This effect has been found experimentally by RMN [55]. In modified silicate the average number of silicon atoms surrounding oxygen atoms decreases from 2 to 1.51 when alkali oxide amount in the system increases. This result is in agreement with experimental result [56], which can explain the transformation of the oxygen from BOs to NBOs one as shown in Fig. 8b. The existing NBOs can be related the depolymerization phenomena mentioned above, which introduces more disorder in the system and consequently increases the entropy of the system influencing probably the glass transition. In this case, we can show the effect of alkali oxides on silicate glasses through the environment of silicon atoms which can be explained by the per-atom potential energy. The per-atom potential energy for silicon atoms is due to its interaction with all other atoms in the simulation especially BOs and NBOs as shown in Fig. 9. The silicon potential energy presents a distribution that is shifted to lower values for alkali silicate glasses, indicating that the presence of alkali cations in the silicate glasses makes the silicon environment more cohesive than the silica glass one. This shift is more important in the case of potassium followed by sodium and finally lithium. For higher content (X N 15 mol%) the force constant of M\\O bond plays a major role on elasticity of the silicate glasses as was indicated by Vaills

Table 3 Glass transition temperature of silicate system and silicate system modified by three alkali oxides.

Tg (K)

(K2O)33(SiO2)67

(Na2O)33(SiO2)67

(Li2O)33(SiO2)67

(SiO2)

1540 ± 10

1240 ± 10

1140 ± 10

2160 ± 10


Fig. 5. Glass transition temperature as a function of molar fraction X for the three alkali oxide modifiers in the silicate matrix. The X values taken in our simulation vary from 0 to 33%. The used cooling rate is equal to 1 K/ps.

et al. [49]. The estimated values of interatomic distance and coordination number from the Si\\Si pair correlation function for all studied systems show that our results are in agreement with experimental data [57]. We also notice that these values are not influenced by the kind or the amount of alkali oxides. The O\\O pair correlation function, called

21

inter-polyhedral distance, shows also a good agreement with experimental results for all the alkali-silicate systems. However, we found a small difference in the O\\O coordination number (11 in our simulation and 13 in experimental results), which may be related to the cut-off distance used in our simulation to integrate over the first shell surrounding an atom. The addition of the alkali oxides does not change the positions of the peaks involving the network forming atoms (Si\\Si, O\\O). Greaves et al. [58] did observe an increase in the Si\\O bond length when different alkali cations were added to glasses; these results were also found by Henderson [59]. The coordination number of potassium in alkali silicate system is higher than the other alkali (sodium and lithium). This result gives us some information about the alkali environments in the alkali silicate glasses indicating that the structures of the three compounds are not the same. This observed difference, is also reported by both experimental techniques and molecular dynamics [56,60], which constitutes a validation of our obtained results. The M\\O pair correlation function shows that the interatomic distance between oxygen and alkali atoms and the coordination number increase respectively in this order: LiO b Na-O b K-O as has been found by Jincheng et al. [4] and in experimental results as shown in Table 4. This could be due to potassium ions having higher percentage of BOs in the first coordination shell of potassium ions [4], which marks that potassium atoms are more homogeneously distributed in the silicate glasses network than sodium and

Fig. 6. Total pair distribution functions for four silicate systems (Sodium silicate glass, Lithium silicate glass, Potassium silicate glass and silica glass) at two temperatures: T = 300 K according to glassy state and T = 3000 K for liquid state.

22


lithium atoms. This effect has been observed experimentally by Charles [61]. The experimental results [57] and the present work shown in Table 4 present some differences especially about the coordination number of potassium and oxygen where coordination numbers are larger in

experimental results. Even though these coordination numbers from molecular dynamics are in agreement with others experimental results [57]. The higher coordination number from experimental result can be explained by instability of the system. Also we notice that all M\\O

Fig. 7. Si\ \O, Si\ \Si, O\ \O and M\ \O partial pair distribution functions obtained by simulating four glass systems (sodium silicate glass, lithium silicate glass, potassium silicate glass and silica glass) at T = 300 K where M = (Li, Na and K). The first peaks correspond to available experimental bond distances [59].


23

Table 4 Parameters of short-range order in alkali silicates estimated from simulated partial pair distribution functions and compared with intensity data of X-ray and neutron diffraction at T = 300 K. X-raya

In this work Structure (SiO2)

(Na2O)33(SiO2)67

(Li2O)33(SiO2)67

(K2O)33(SiO2)67

Neutrona

i-j pair

ri−j(Å)

Ni−j

ri−j(Å)

Ni−j

ri−j(Å)

Ni−j

Si-O O-O Si-Si Si-O O-O Si-Si Na-O Si-O O-O Si-Si Li-O Si-O O-O Si-Si K-O

1.63 2.71 3.29 1.61 2.66 3.15 2.32 1.61 2.67 3.02 2.07 1.62 2.66 3.15 2.67

4.00 6.00 4.00 4.00 5.20 4.05 5.00 4.00 4.88 4.20 4.90 3.98 5.45 3.15 7.06

1.62 2.65 3.11 1.62 2.65 3.21 2.36 1.62 2.65 3.13 2.07 1.62 2.65 3.23 2.65

3.90 5.50 3.90 4.00 5.20 3.60 5.80 3.70 5.60 3.80 3.80 3.80 13.20 3.50 13.20

1.62 2.66 3.10 1.61 2.65 3.19 2.34 1.62 2.65 3.15 2.07 1.61 2.65 3.22 2.65

4.10 6.20 4.40 4.30 6.20 4.40 6.10 4.20 6.30 4.50 3.90 4.20 12.40 4.60 12.40

Ni−j average coordination number ±0.3, ri−j average distance ±0.01. a Experimental results carried out by Waseda et al [57].

bonds are longer than Si\\O in alkali silicate glasses. This result is already found in other works for sodium silicate glass [62]. The Na\\O coordination number NNa-O(r) curves shows an inflection of about NNaO(r) ~ 5 that indicates the number of oxygen atoms surrounding Na atoms. This fivefold coordination agrees well with the XAFS data [63]. On the other hand we remark that the first peak of O\\O and M\\O are in coincidence in potassium silicate glass as was found by Hannon et al. [56], this phenomena disappeared when M becomes small as shown in Fig. 7. The partial radial function for M\\M computed from our simulation is reported as the broad peak in the silicate glasses for all cases as shown in Fig. 8(a), where M = K,Na or Li. This interatomic distance increases in this order: Li-Li b Na-Na b K-K as found also by du et al. [4].

Two effects are used in this work concerning the glass transition. The first one is based on the effect of cooling rate on the glass transition shown that this later varies proportionally with the cooling rate. This dependence of Tg on the cooling rate has been shown by many works [22,26,1]. This phenomenon leads us to suggest that low cooling rate makes the glassy material more stable than high cooling rate [31]. In the silica glassy materials, a strong glass former, reveal that the

5. Discussion Overall, if we restrict ourselves to the thermodynamic and structural properties of the silica and alkali silicate glasses as function of alkali oxides molar fraction (X%), Habasaki potential and ours simulation, appear to offer a good agreement with experimental behavior and values measured from experimental results. The comparison between experimental results and our simulation about the interatomic distance and coordination numbers is hopeful, especially for the silica, the sodium silicate and lithium silicate glasses. On the other hand, the Table 4 summarizes the interatomic distances and coordination numbers of the simulated glasses found in this work, together with those obtained by Du et al. [4] using MD simulation. The small difference between the two results can be attributed to the difference in the used interatomic potential. A decrease in the Si\\O bond length in alkali silicate network, as observed from our simulation results and available experimental data (Table 4), could be explained by the decrease in the Si-BOs contribution [4]. One consequence of the increase of the cohesiveness with its environment is related to the decrease of the Si\\O and Si\\Si bond length. In potassium silicate glasses the number of oxygen atoms around oxygen and potassium atoms obtained in our simulations shows some differences with experimental results. This discrepancy can be explained by the instability of the potassium silicate glass where the potassium ions in this case like to be homogeneously distributed in the silicon–oxygen network [4]. The interatomic distance K\\O is close to that of O\\O, can be explained by the fact that ionic radii of both oxygen and potassium are quasi equal, this implies that the substation between each other produced no effect on the whole structure.

Fig. 8. a) K\ \K, Na\ \Na, Li\ \Li partial pair distribution functions obtained by simulating four glass systems (sodium silicate glass, lithium silicate glass, potassium silicate glass and silica glass) at T = 300 K where M = (Li, Na and K). b) O\ \Si cumulative coordination number of the (SiO2) (67) (M2O) 0.33 glasses obtained from the classical MD simulation, where M = K, Na, Li.

24


Fig. 9. Potential energy distribution of silicon atoms in alkali silicate (green curve) and in silica glass (red curve) for the three modifier cases at T = 300 K. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Stokes–Einstein relation holds all the way down to the calorimetric Tg. On the other hand, it is shown for fragile liquids. In many works, Tg had been related to change in the liquid structure [64]. The glass transition temperatures, in which liquids fall out of internal equilibrium, furnish rather direct report on the magnitude of interparticle interactions and therefrom are highly useful to diagnose the chemical constitution effect on liquid properties [65]. In our work, the fictive temperature concept was extracted through the enthalpy variation during the material quenching from melting state. The fictive temperature is a pertinent parameter to characterize any non-equilibrium state and this behavior is described in terms of this concept, which differs from the equilibrium temperature T but relaxes toward it as the system ages [66]. The second effect on Tg is its dependence on alkali ions type and content. As shown above, Tg decreases in the presence of alkali oxides in the silica network. Many previous works were interested in studying the glass transition temperature as effect of the structure changes in the liquid state. In this work, we are interested in showing the effect of glass transition on structure of the glassy state. So the values of Tg in multicomponent glasses depend upon the measurement conditions and glass structure, i.e. environment of species, their variation with composition, temperature and pressure [10]. The aim is to try explaining this phenomenon

Fig. 10. Molar density of silicon atoms dsi (mol cm−3) as a function of molar fraction X for the three alkali oxide modifiers in the silicate matrix. The X values taken in our simulation vary from 0 to 33% at T = 300 K.

by some effects such as the molar density of silicon atoms dsi. For that, we computed dsi as a function of M2O content, where M = Li, Na, K. This density is determined as a combination between M2O content, mass density of (SiO2)(1-X) (M2O)x glasses and the molar weights for both sodium oxide and silicon oxide [49], it can be written as: X M1 dSi ¼ ρ= M0 þ 1−X

ð3Þ

where ρis the mass density of (SiO2) (1-X) (M2O)X glasses. M0 and M1 are the weight of SiO2 and M2O, respectively. The computed dsi are shown in Fig. 10 as function of M2O content for the three cases. The results are in agreement with experimental results [49]. Therefore, we can link the decrease of glass transition as function of X mol% M2O content with decreasing of the molar density of silicon atoms dsi. On the other hand, we can discuss the decrease of the glass transition by the depolymerization of the silica network where the BOs and NBOs play a major role in the global structure of the alkali-silicate system. The glass transition phenomenon occurs due to the increasing viscosity of overcooled liquids, so Tg strongly depends on the polymerization ratio of the network [42]. To illustrate this suggestion, we extend our analysis to longrange order by dividing the tree-dimensional simulation box into twodimensional slices as shown in Fig. 11. We observe that the alkali atoms in the case of NBOs organize them-selves in forms of long chains or channels. These channels separate the total silica network to many silicate sub-networks, which are surrounded by NBOs. These sub-lattice are connected between each other by an ionic bonded sub-lattice (percolated channels) where NBOs are considered as an interface between different covalent bonded SiO2 networks in the alkali-silicate glass. These alkali ions can migrate along (percolated) alkali-rich channels [67–70]. In an alkali silicate glass, the liquid state is characterized by a mobility of the alkali ions higher by several orders of magnitude than that of the ions of the silica network. Several scenarios have been proposed to explain such a difference. One of these scenarios suggest the existence of preferential channels diffusion of alkaline ions in the matrix of Si\\O [28,34,71–72]. These systems show a scale characterizing the medium-range order channels in which are confined the alkali ions. The model (the modified random network-MRN) proposed by Greaves is a good description of these structures [73–74]. The alkali ions partially break the SiO2 network structure and in the case of Na and Li, they not only modify the structure on local length scales but also introduce new features at long length scales. These features manifest them self


25

Fig. 11. Thin slices showing the structures of (M2O)33(SiO2)67 glasses at T = 300 K with M= Li, Na or K. One can easily see that alkali modifying cations form a connecting silica glass network. Covalent bonds are shown by the red-blue lines and ionic bonds by the yellow-blue lines. This 2-dimension cut is taken from the simulation box as slice using OVITO [20]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

by channels which can be explained by an increasing of alkali ions mobility corresponding to a decrease of the activating energy. The MNR model assumes as a result of the coordination inferred from the EXAFS studies that the network modifier cations are not homogeneously distributed. The existence of long ionic bonded sub-lattice formed frozen clusters regions, which can affect the glass transition. We can relate the above results about alkali environment and separation phenomena shown in Fig. 11, in which the homogeneous distribution of potassium atoms already indicated by higher coordination number of potassium in silicate to a lower phase separation in potassium silicate glasses than sodium and lithium silicate glasses [60]. Thus, potassium does not tend to segregate or to form clusters, but it is homogeneously distributed in the network. On the other hand, the formation of glassy state during the quenching can be easier shown for homogeneous distribution than inhomogeneous distribution one, which can be related to the occurrence of glassy state in the case of potassium silicate glass earlier than sodium and lithium silicate glasses as shown above.

6. Conclusion and outlook We have simulated the silica and alkali-silicate glass by mean of molecular dynamics. The non-equilibrium state was characterized by fictive temperature and its dependence on the effect of cooling rate on the glass transition. We are interested also in studying the opposite effects of alkali on both Tg and structural properties of silicate binary glasses. The results of our molecular dynamics simulation are in good agreement with structural results obtained experimentally by X-ray, EXAFS and neutrons diffraction. This validates our simulation based on the potential model proposed by Habasaki. In fact, Tg is linked directly to the change of local structural, as well as, the transformation of the oxygen atoms from BOs to NBOs oxygen is the effect responsible of this change. The atomic visualization showed that the NBOs organize them-self in forms of long chains or channels. These channels separate the total silica network to many silica sub-networks, which are surrounded by NBOs. This effect was clearly shown in the cases of sodium and lithium silicate glasses. However these channels were not observed for potassium which has an ionic radius close to that of oxygen leading to a homogenous structure. The next step will be the study of the coupling between optical and mechanical properties of silicate glass with other kinds of modifiers such as Mg and Ca, in order to link simulation results to experimental techniques like Brillouin and Rayleigh spectroscopy.

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