Screening dependence of the dynamical and structural properties of BKS silica
A. Kerrache, V. Teboul
*, A. Monteil
Laboratoire POMA, UMR CNRS 6136, Universite´ dÕAngers, 2 boulevard Lavoisier, 49045 Angers, France Received 23 March 2005; accepted 28 July 2005
Available online 1 September 2005
Abstract
Molecular dynamics simulations of amorphous silica are carried out on a large temperature range using a modified version of the BKS inter-atomic potential. We investigate the dependence on the screening procedure of the structural and dynamical properties of amorphous silica. We show that an increased screening of the electrostatic interaction leads to a decrease of the diffusion constants and then to better agreement with experimental data, while structural properties are unchanged. We show that the Arrhenius depen- dence of the diffusion constants may be reproduced in this case up to a temperature of 4000 K with activation energies very similar to the experimental data.
Ó2005 Elsevier B.V. All rights reserved.
PACS: 64.70. Pf; 61.20. Lc
Keywords: Glass-transition; Silica; Potential; Strong glass-former
1. Introduction
Silica is one of the most extensively studied materials in condensed matter physics, chemistry, material sci- ence, and engineering. Silica is important in various do- mains ranging from geophysics to the technology of optical fibers. For the glass-transition problem silica is of particular importance as the main prototype of strong glass-formers[1,2]. The origin of the Arrhenius temper- ature dependence of the viscosity of some liquids like silica while the viscosity of others liquids follows a super-Arrhenius law is still a matter of debate. As the molecular dynamics (MD) simulation method has been found to reproduce well the non-Arrhenius character of some glass-formers[3]as well as the Arrhenius char- acter of others [4,5], this method seems to be well adapted [6] for the study of the origin of the fragility.
The Arrhenius or non-Arrhenius character of glass- formers appears in MD simulations as a direct conse- quence of the intermolecular potential choice. A number of interaction potentials have been proposed for silica [4,7–9]. Most of these potentials reproduce the structural properties of silica [4]. However dynamical properties are usually poorly reproduced[4]. Some of these poten- tials [9]use enhanced screening procedures to take into account the infinite range of the electrostatic interac- tions while others like the BKS potential [7] use the more precise Ewald summation technique [10]. The po- tential proposed by BKS[7]is one of the most popular inter-atomic potential as long as dynamical properties are concerned. This two-body potential has proved to well reproduce the structural properties of silica [4]
and leads to dynamical properties that are among the most compatible with experimental data[4,5]. The diffu- sion coefficients obtained with this potential are however still too high to be exactly compatible with the experi- mental data[4,11,12].
0301-0104/$ - see front matter Ó2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2005.07.039
* Corresponding author. Tel.: +33241735004; fax: +33241735216.
E-mail address:victor.teboul@univ-angers.fr(V. Teboul).
www.elsevier.com/locate/chemphys
In this article, we study the effect of a modification of the screening procedure of the BKS potential on the dynamical and structural properties of silica. For this purpose, we replace the long range electrostatic interac- tion described with Ewald summation technique with the reaction field (RF) method [10]. The reaction field method considers that for distances sufficiently large (larger than a defined radius R) the positions of the charges may be replaced by a continuum of dielectric constant e. The reaction field method increases the screening on the electrostatic interaction when the cavity radius R decreases. This method will then allow us to tune the screening on the electrostatic interaction through the choice of two parameters: the dielectric con- stanteand the radiusRof the dielectric cavity. Because the dielectric constant is a known physical data we do not have much tuning possibility with e, we will then concentrate on a modification of the radiusRof the cav- ity. Using the reaction field method instead of the Ewald technique is equivalent to a modification of the screen- ing at long range. This method leads to a screening near- est to the screening of other potentials[9]. This method may be well adapted for amorphous or liquid silica.
We show in this article that the use of the RF method with a 10 A˚ cut-off radius leads to the same structure than the original BKS potential[7]. The main dynamical effect is a slow down of the dynamics. This slow down leads to diffusion coefficients that are in better agree- ment with experimental data than the original potential.
We show that the Arrhenius dependence of the diffusion coefficients with temperature is reproduced with activa- tion energies for oxygen and silicon atoms that are com- patible with the experimental energies. This Arrhenius dependence begins here at a temperature of 4000 K in- stead of 3330 K for the original BKS potential. This modification of the BKS potential is then of particular interest for the study of silica as a prototype of strong glass-formers.
2. Calculation
The present simulations were carried out for a system of 9000 particles (3000 Si + 6000 O) for the lowest tem- peratures studied (2900, 3000, 3100 and 3250 K), 4608 particles (3500 K) or 1125 particles for the highest tem- peratures investigated (4000–8000 K). The calculations used the molecular dynamics (MD) method with a Ver- let Algorithm to integrate the equations of motion. The time step was chosen equal to 1015s. We have used the modified BKS potential, which is to our knowledge one of the best existing potentials to study the dynamics of silica. A very short range repulsive part was added, as described by Guissani and Guillot[13], in order to elim- inate the possible short range divergence of this poten- tial. The reaction field method was used with various
cut-off radii Rcto calculate the long range electrostatic part of the potential. The dielectric constant of silica being relatively large we have used an infinite dielectric constant in this calculation. The modified interaction potential has then the form:
Vij ¼qiqj rij 1þ1
2Br3ij R3c
!
þAijebijrijCij r6ij þ4eij
rij
rij
24
rij
rij
6!
. ð1Þ
We then subtract the constant valueVij(Rc) from the po- tential in order to eliminate the discontinuity of the function at the cut-off. The different coefficients values are listed in Table 1. B¼2ðe1Þð2eþ1Þ where e is the relative dielectric constant. With e=1, B is then equal to 1.
The reaction field method then replace simply, if B= 1, the Coulombic force with
fij¼qiqj
r3ij 1r3ij R3c
!
rij forrij<Rc andfij¼0 forrij>Rc.
ð2Þ The density was set constant at 2.3 g/cm3. The box size is then constant at 50 A˚ . The system is heated at a tem- perature of 7000 K to insure homogenization. It is then cooled to the different temperatures of study using a Berendsen thermostat [14]. The simulations are then aged during 20 ns in order to insure stabilization before any treatment. The simulations are performed at con- stant temperature in the (N,V,T) ensemble. However, the total energy of the system has been found to be very stable in the simulations.
In our simulations, we use different cut-off valuesRc
in order to study the screening dependence of the struc- ture and dynamics of liquid silica. However, this change of the cut-off Rc has an effect on the pressure of the li- quid. Because the pressure modification may result in a modification of the diffusion we will investigate this ef- fect before any other calculation. In molecular dynamics simulations, the pressure is usually obtained from the formula
P ¼qkBT þ 1 3V
X
i<j
fijrij
* +
ð3Þ
Table 1
Force-field parameters used in this work
Aij(eV) bij(A˚1) Cij(eV A˚6) rij(A˚ ) eij(kJ/mol)
O–O 1388.7730 2.76 000 175.0 2.2 0.0344
Si–O 18003.7572 4.87318 133.5381 1.31 1.083
Si–Si 0. 0. 0. 0.42 1219.45
qO=1.2 e.
qSi= 2.4 e.
B= 1.
From[7,13].
or equivalently P¼qkBT2
3pX
a;b
qaqb Z 1
0
r3gabðrÞdvabðrÞ
dr dr ð4Þ
with a, b = O or Si, vab(r) is the potential described above, gab(r) the radial distribution function and q is the density. We will show later that the radial distribu- tion functions are only slightly modified by the cut-off.
In this formula, the modification of the electrostatic part of the potential is then the main origin of the pressure modification. We then display inFig. 1the electrostatic pressure evolution with the cut-off Rc: PelecðRPcelecÞPð1Þelecð1Þ in percents, for two different temperatures (3100 and 6600 K). We observe an oscillation of the pressure around its value forRc=1. This oscillation is due to the oscillations of the radial distribution functions.
The oscillation decreases rapidly due to the reaction field modification of the electrostatic force. The two different temperatures show roughly the same oscillations, the pressure modification being however larger for the low- est temperature. The pressure modification decreases more rapidly at the higher temperature (6600 K). This modification disappears for a cut-off larger than 20 A˚ at 3100 K and 12 A˚ at 6600 K. We also see in this figure that for some particular cut-off values the pressure is only slightly modified by the reaction field method.
3. Results
This section is organized in two parts: in a first part we investigate the screening dependence on the struc- tural properties of Silica. We investigate for this purpose the radial and angular distribution functions and show that they are not affected by the screening procedure used. In a second part, we investigate the modification of the dynamical properties. We calculate the diffusion coefficients and show that they are weaker than the coef- ficients calculated with the original BKS potential lead- ing to better agreement with experiment if we suppose that the liquid has an Arrhenius temperature depen- dence in the temperature range in between experimental and simulation data. We also show that the activation energies calculated with this procedure are approxi- mately equal to the experimental activation energies.
We show inFig. 2(a)–(c) the radial distribution func- tions between, respectively, oxygen atoms, silicon atoms, and between silicon and oxygen, calculated with a 10 A˚ cut-off radius for three typical temperatures. The
-40 -30 -20 -10 0 10 20 30 40
0 5 10 15 20 25
Pressure deviation (percent)
Cutoff radius (Angstrom)
Fig. 1. Electrostatic pressure evolution with the cut-off Rc:
PelecðRcÞPelecð1Þ
Pelecð1Þ in percents, for two different temperatures: 3100 K (continuous line) and 6600 K (dashed line). The small line shows the pressure at the particular cut-off valueRc= 10 A˚ .
0 0.5 1 1.5 2 2.5 3 3.5
0 2 4 6 8 10 12
g(r)Si-Si
Si-Si
3250 K 4400 K 6600 K
0 2 4 6 8 10 12
0 2 4 6 8 10 12
g(r)Si-O
Si-O
3250 K 4400 K 6600 K
0 0.5 1 1.5 2 2.5 3 3.5
0 2 4 6 8 10 12
g(r)O-O
Distance R ( °A)
O-O 3250 K
4400 K 6600 K a
b
c
Fig. 2. Radial distribution function between: oxygen atoms (a), silicon atoms (b), oxygen and silicon atoms (c) at three different temperatures:
3250 K (continuous line), 4400 K (dashed line), 6600 K (dotted dashed line).
positions of the first peaks of the radial distribution functions are shown inTable 2, and compared to previ- ous calculations obtained with different potentials and experimental values at low temperature. A good agree- ment is observed between the different values. In partic- ular, the results obtained with the reaction field method and the Ewald method (indicated with R infinite in Table 2) are in good agreement. The modification of the cut-off radius (i.e., the screening intensity), inside the interval 8 A˚ to infinite, has been found to affect only slightly the radial distribution function. The different intermolecular potentials compared in Table 2 repro- duce satisfactorily the experimental positions of the first peaks of the radial distribution functions. The difference between the experimental and simulated values is slight for the simulations using the RF method and the BKS potential. We show in Fig. 3 the angular distribution functions Si–O–Si and O–Si–O, which with the radial distribution functions, determinate the silica network structure. The results agree well with previous simula- tions and experimental data, as seen in Table 3. The dependence of the structure as described by the radial distribution functions and the angular distribution func- tions is then slightly dependent on the cut-off radius of the reaction field method (i.e., on the screening intensity).
In contrast, the dynamics is deeply affected by the choice of the long distance screening procedure. Table 4shows that an increase of the screening (i.e., a shorter cut-off radius) leads to a decrease of the diffusion coeffi- cient. In other words, an increased screening leads to a slowdown of the dynamics. Adjusting the screening ra- dius it is then possible to approach the diffusion coeffi- cients deduced from experimental data. Fig. 4 shows that this modification of the screening, does not change the pure exponential dependence (Arrhenius depen-
Table 2
Location of the first maximum of the radial distribution function, determined from this work and compared to other simulations and experimental data
Simulations This work Experiment
O–O 2.59 (Rinfinite)[5] 2.61 (R= 10 A˚ ) 2.626[18]
2.62[8] 2.65[19]
2.66[15]
2.65[16]
2.60[17]
Si–O 1.595 (Rinfinite)[5] 1.60 (R= 10 A˚ ) 1.608[18]
1.63[8] 1.62[19]
1.62[15]
1.61[16,17]
Si–Si 3.155 (Rinfinite)[5] 3.11 (R= 10 A˚ ) 3.12[19]
3.16[8] 3.077[20]
3.08[15]
3.13[16]
3.14[17]
0 0.5 1 1.5 2 2.5
60 80 100 120 140 160 180
Bond-Angle distribution
O-Si-O 2900 3500 4000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
60 80 100 120 140 160 180
Bond-Angle distribution
Angles (degrees) Si-O-Si
2900 3500 4000
b a
Fig. 3. Angular distribution functions (Si–O–Si and O–Si–O) at three different temperatures: 2900 K (continuous line), 3500 K (dashed line), 4000 K (dotted dashed line).
Table 3
Location of the maximum of O–Si–O and Si–O–Si angles determined from this work and compared to other simulations and experimental data
Angles (degrees)
Simulations This work Experiment
O–Si–O 108.3 (Rinfinite)[5] 107.3 (R= 10 A˚ ) 109.5[19]
109[15] 109.47[21]
109.7[22]
109.4[23]
Si–O–Si 152 (Rinfinite)[5] 147.3 (R= 10 A˚ ) 144[19,22]
145[15] 142[21,24]
153[23]
Table 4
Diffusion coefficients versus cut-off radius for oxygen and silicon atoms at a temperature of 3310 K
Rc(A˚ ) DO(1011m2/s) DSi(1011m2/s)
8.0 2.56 1.27
10.0 3.02 1.72
12.5 3.17 1.64
Infinite[25] 7.40 4.40
dence) of the diffusion coefficient with temperature.
Moreover this Arrhenius dependence appears at higher temperature when the screening is increased. For a cut-off radiusR= 10 A˚ , we observe an Arrhenius depen- dence of the diffusion coefficient which begins around 4000 K instead of 3330 K forRinfinite [5]. The Arrhe- nius character of supercooled silica is then dependent on long distances interactions. The modification of the cut-off radius changes also slightly the activation ener- gies associated to the diffusion of oxygen and silicon.
Table 5shows that this slight modification leads how- ever to a better agreement with the experimental values.
In order to be able to compare the experimental and simulated diffusion coefficients compatibility, we display inFig. 4the silicon and oxygen diffusion coefficients for a larger scale ranging from 300 to 7000 K. The lines cor- respond in this figure to a pure Arrhenius dependence of the diffusion coefficients. For a 10 A˚ cut-off radius we observe that the simulated oxygen diffusion coefficients
are compatible with the experimental data of Mikkelsen [11]and an Arrhenius evolution at low temperature. For silicon atoms, we observe a rough compatibility with the results of Brebec et al.[12]. We observe inFig. 4that the diffusion coefficients obtained with the increased screen- ing procedure are then in better agreement with diffusion coefficients deduced from the experimental data and an Arrhenius behavior than the original BKS potential using the Ewald technique or an infinite cut-off radius.
4. Conclusion
In this work, we have used molecular dynamic simu- lations to investigate the screening dependence of the structural and dynamical properties of silica with the BKS potential. We have shown that the use of an in- creased screening obtained with the reaction field meth- od with a 10 A˚ cut-off radius leads to the same structure than the original BKS potential and to a slowdown of the dynamics. This slowdown leads to diffusion coeffi- cients that are in better agreement with experimental data than the original potential. We have shown that the Arrhenius dependence of the diffusion coefficients with temperature is reproduced. This Arrhenius depen- dence begins here at a temperature of 4000 K instead of 3330 K for the original BKS potential. This modifica- tion of the BKS potential may then be of particular
Table 5
Activation energies corresponding to an Arrhenius fit D=D0exp(EA/kBT) of the oxygen or silicon diffusion coefficients evolution with temperature
Activation energyEA(eV)
Simulation This work Experiment
O 4.66 (Rinfinite)[5] 4.72 (R= 10A˚ ) 4.7[11]
Si 5.18 (Rinfinite)[5] 5.36 (R= 10A˚ ) 6.0[12]
1.0E-22 1.0E-20 1.0E-18 1.0E-16 1.0E-14 1.0E-12 1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02
1 2 3 4 5 6 7 8
D (O and Si) (cm2 /s)
104/T (K-1)
Mikkelsen
Brebec et al.
EA = 4.7 eV
EA = 6 eV Si O
1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 1.0E-04
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 EA(O) = 4.72 eV
EA(Si) = 5.36 eV
Fig. 4. Oxygen (squares) and silicon (circles) diffusion coefficients versus 1/T. The experimental diffusion coefficients low temperature values are plotted for oxygen atoms[11](continuous line) or silicon atoms[12](crosses). The lines (continuous for oxygen atoms or dashed for silicon atoms) correspond to a fit with an Arrhenius law of the silicon or oxygen diffusion coefficients with temperature. Error bars plotted in the right end side of the figure display the largest errors which may arise from different fits of the high temperatures simulations. The partial lines (bold continuous for oxygen atoms or bold dotted for silicon atoms) correspond to an Arrhenius extrapolation of simulations obtained with the BKS potential using the Ewald technique (Rinfinite with our notations). The dotted line corresponds to the Arrhenius interpolation of the experimental silicon diffusion coefficients to higher temperatures. Inset: Oxygen (squares) and silicon (circles) diffusion coefficients versus 1/T. The lines (continuous for oxygen atoms or dashed for silicon atoms) correspond to a fit with an Arrhenius law of the silicon or oxygen diffusion coefficients with temperature.
interest for the study of silica as a prototype of strong glass-formers.
References
[1] C.A. Angell, J. Non-Cryst. Solids 131–133 (1991) 13.
[2] (a) A. Kerrache, V. Teboul, D. Guichaoua, A. Monteil, J. Non- Cryst. Sol. 322 (2003) 41;
(b) V. Teboul, A. Monteil, L.C. Fai, A. Kerrache, S. Maabou, Eur. Phys. J. B 40 (2004) 49.
[3] (a) V. Teboul, C. Alba-Simionesco, J. Phys.: Cond. Mat. 14 (2002) 5699;
(b) V. Teboul, S. Maabou, L.C. Fai, A. Monteil, Eur. Phys. J. B 43 (2005) 355.
[4] M. Hemmati, C.A. Angell, in: M. Aoki, R. Hemley (Eds.), Physics Meets Geology, Cambridge University Press, 1998.
[5] (a) W. Kob, J. Phys.: Cond. Mat. 11 (1999) R85;
(b) K. Vollmayr, W. Kob, K. Binder, Phys. Rev. B 54 (1996) 5808;
(c) J. Horbach, W. Kob, K. Binder, C.A. Angell, Phys. Rev. E 54 (1996) R5897;
(d) J. Horbach, W. Kob, Phys. Rev. B 60 (1999) 3169.
[6] S. Chaussedent, V. Teboul, A. Monteil, Curr. Opin. Sol. State Mat. Sci. 7 (2003) 111.
[7] B.W.H. Van Beest, G.J. Kramer, R.A. Van Santen, Phys. Rev.
Lett. 64 (1990) 1955.
[8] S. Tsuneyuki, M. Tsukada, M. Aoki, M. Matsui, Phys. Rev. Lett.
61 (1988) 869.
[9] (a) J.D. Kubicki, A.C. Lasaga, Am. Miner. 73 (1988) 941;
(b) Y. Matsui, Phys. Chem. Miner. 169 (1988) 234;
(c) L. Van Brutzel, E. Bouchaud, Physics of glasses, in: P. Jund, R.
Julien (Eds.), AIP Conference Proceedings 489, New York, 1999, p. 231;
(d) B.P. Feuston, S.H. Garofalini, J. Chem. Phys. 89 (1988) 5818.
[10] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, New York, 1990.
[11] J.C. Mikkelsen, Appl. Phys. Lett. 45 (1984) 1187.
[12] G. Brebec, R. Seguin, C. Sella, J. Bevenot, J.C. Martin, Acta.
Metall. 28 (1980) 327.
[13] Y. Guissani, B. Guillot, J. Chem. Phys. 104 (1996) 7633.
[14] H.J.C. Berendsen, J.P.M. Postma, W. Van Gunsteren, A. Di Nola, J.R. Haak, J. Chem. Phys. 81 (1984) 3684.
[15] M. Benoit, S. Ispas, P. Jund, R. Julien, Eur. Phys. J. B 13 (2000) 631.
[16] P. Vashishta, R.K. Kalia, J.P. Rino, Phys. Rev. B 41 (1990) 12197.
[17] M. Wilson, P.A. Madden, M. Hemmati, C.A. Angell, Phys. Rev.
Lett. 77 (1996) 4023.
[18] D.I. Grimley, A.C. Wright, R.N. Sinclair, J. Non-Cryst. Sol. 119 (1990) 49.
[19] R.L. Mozzi, B.E. Warren, J. Appl. Crystal. 2 (1969) 164.
[20] J.H. Konnert, J. Karle, Acta Crystallogr. A 29 (1973) 702.
[21] E. Dupree, R.F. Pettifer, Nature 308 (1984) 253.
[22] P.G. Coombs, J.F. De Natale, P.J. Hood, E.K. Mc Elfresh, R.S.
Wortman, J.F. Schachelford, Phil. Mag. 51 (1985) 39.
[23] J.R.G. Da Silva, D.G. Pinatti, C.E. Anderson, M.L. Rudee, Phil.
Mag. 31 (1975) 713.
[24] R.F. Pettifer, R. Dupree, I. Farnan, U. Sternberg, J. Non-Cryst.
Sol. 106 (1988) 408.
[25] Y. Zhang, G. Guo, K. Refson, Y. Zhao, J. Phys.: Cond. Mat. 16 (2004) 9127.
