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VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 Simulation of Diffusion Mechanism and Heterogeneous Dynamics in Network-forming Liquid Nguyen Thi Thanh Ha1,*, Nguyen Anh Dung2, Nguyen Quang Bau3, Pham Khac Hung1 1 Department of Computational Physics, Hanoi University of Science and Technology, Vietnam 2 Facutly of Electrical Engineering Technolog, Hanoi University of Industry 3 Department of Physics, Hanoi National University, Hanoi, Vietnam Received 18 November 2016 Revised 16 December 2016; Accepted 28 December 2016 Abtract: We have studied the diffusion mechanism in network-forming liquids via molecular dynamics simulation. Two models of silica liquid at temperatures of 2600 and 3500 K have been constructed. The motion of atoms is analyzed through reactions SiOxSiOx1, OSiyOSiy1 and coordination cells. It was shown that the reactions happen not randomly in the space, but they occur more frequently with some coordination cells and bonds Si-O. This effect enhances upon lowering the temperature. The number of reactions happening in the coordination cells is strongly correlated with the mobility of CC atom. We found that there are large rigid clusters which compose of immobile atoms and exist for long time. Further, the dynamics heterogeneity (DH) is determined via sets of: (1) most mobile or immobile atoms; (2) atoms belonging to large rigid clusters. We show that the percolation rigid cluster responds to the dynamical slowing down present near glass-transition. Keywords: Simulation, silica, liquid, dynamic heterogeneity, rigid cluster. 1. Introduction The glass-forming liquids, when cooled down below melting point fast enough, will form a glass at the temperature Tg called the glass-transition temperature. Close to the glass-transition, the structure of these liquids appear to change subtly but dynamical properties e.g. the structural relaxation time, the diffusion constant, and the viscosity increase by orders of magnitudes [1-3]. The reason for the dynamical slowing down is still not properly understood and represents a major challenge in the condensed matter. Moreover, the liquid near glass transition exhibits the dynamics heterogeneity (DH) [4-6]. The examination of particle motion in a dense colloidal suspension [7-8] reveals that close to the glass-transition, particle motion is strongly spatially correlated. Recent simulations studied the _______  Corresponding author. Tel.: 84-983012387 Email: ha.nguyenthithanh1@hust.edu.vn 32 N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 33 correlation functions for Lennard-Jones systems and finding the evidence for the DH [9-10]. In other works [11-13] one visualized the set of most mobile and immobile particles and showed that they form large clusters which migrate in the space over times. A group of particles follows one another along string-like paths. The particle mobility seems to be a result of the dynamics facilitation [14]. In previous work [15] we study reactions SiOxSiOx1 and OSiy OSiy1 which happen in the liquid and govern the diffusion. We show that the diffusion constant depends not only on the rate of reactions, but also on the spatial distribution of reactions. It is interesting to determine how the reactions affect the DH and the dynamical slowing down. Moreover, the clustering of mobile or immobile particles may be the result of specific distribution of reactions in the space. However, this is not clarified yet and motivates us to perform a detail numerical study in this direction. 2. Computational procedure MD simulation is carried out for the liquid model composing 666 silicon and 1332 oxygen atoms. We used the van Beest–Kramer–van Santen (BKS) potential [16]. The MD step is equal to 0.47 fs. Initial configuration is generated by randomly placing all particles in the simulation box. Then this configuration is heated to 5000 K and cooled down to the specified temperature. After that, the sample has been relaxed in N-E-V ensemble (the constant volume and energy) until the system reached the equilibrium. The pressure is kept to zero. We have prepared two liquid models at 2600 and 3500 K. More details about the procedure of preparing BKS model can be found elsewhere [17]. The diffusion mechanism is analyzed through the coordination cell and reaction. The coordination cell consists of a central atom (CC atom) and neighbors. The CC atom (Si or O) is linked with neighbors via bonds Si-O. The cutoff distance used to determine the coordination cell is 2.28 Å. We have labeled the coordination cell by the label of CC atom. As a reaction happens in the coordination cell, a bond breaks or new bond is created. Most reactions are SiOx  SiOx1 and OSiy  OSiy1. During the simulation we have determined the mean square displacement of CC atom and the number of reactions happening in the coordination cell. In order to clarify DH we determine the cluster of atoms. A cluster is defined as a set of atoms that forms at least a bond with another atom in the set. Here, a bond is created by two atoms Si and O, the distance between which is less than the cutoff distance. The schematic illustration of coordination cell and cluster is shown in Fig. 1 a) b) c) Fig1. The schematic illustration of (a) and (b) the coordination cell SiO x, OSiy, (c) the cluster. Here the black, blue sphere represents silicon and oxygen, respectively. The black line indicates a bond between two atoms. 34 N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 We consider set A1 of N atoms which are chosen randomly from the system. The mobility of atoms of A1 in average is close to maverage (averaging the mobility of all atoms in the system). We also consider the motion of N atoms for time tDH; N is smaller than 15% of all atoms in the system. These atoms are chosen so that their mobility in average strongly deviates from the value maverage. It means that the set of N atoms includes the mobile or immobile atoms. We denote the set of these N atoms to Ak. Two quantities of interest are the number of atoms in the largest cluster MACL and the number of clusters MCL which are found from A1 and Ak. We calculate MACL, MCL of A1 and Ak at 11 moments within time tDH. Then we compare MACL, MCL of Ak with ones of A1. In the case when MCL of A1 is bigger and MACL of A1 is smaller than ones of Ak, the system shows DH. The difference in MACL, MCL indicates the degree of DH. If a bond remains unchanged for time tDH, this bond is called unbroken bond. The cluster of atoms linked via unbroken bonds is called the rigid cluster. The bond may break and then restore after some times. For instance, an oxygen atom leaves the coordination cell SiO4 breaking a bond. Then this oxygen comes back. As a result, the broken bond Si-O is restored. If the bond breaks at t < tDH, and then restores at t = tDH, we call it the restored bond. The DH, as we see below, can be seen by monitoring the motion of atoms of rigid clusters. 3. Results and discussion The number of reactions The constructed models have the structural characteristics closed to previous works result in ref.[17,18]. The result is consistent with the experiment in the positions and heights for peaks of PRDFs. In Table 1 we show the dynamics and structure characteristics. As temperature increases the fraction of tetrahedral units C4 monotonously decreases, meanwhile fractions C5 (SiO5) and Cother (SiO3 and SiO6) increases in considered temperature interval The diffusion constant is calculated from the mean square displacement of atom via Einstein equation. We determine the rate of reaction (ξ) from the time dependence of number of reaction. 100000 3500 K 2600 K 80000 60000 40000 20000 0 0 20000 40000 60000 80000 Steps Fig.2. The number of reactions as a function of n steps. Fig.2 shows the number of reactions for two samples. All points fall well on straight lines. The slopes of these lines are used to determine the rate of reactions. One can see that both diffusion constant and rate of reaction strongly decreases with decreasing the temperature. However, ξ decreases much slower than diffusion constant. This indicates the correlation between reaction and diffusivity but the reactions happen in these samples not by the same way. This effect can be seen from Fig.3. Here mcc is the number of coordination cells in which the reactions happen; munbrb, mresb is the number of unbroken bonds and restored bonds, respectively. If the reactions happen more frequently with some coordination cells or bonds, mcc becomes smaller and munbrb bigger than for the case of homogeneous distribution of reactions. As shown in Fig.3, when the same number of reactions N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 35 happens in both samples, mcc is smaller for low-temperature sample and munbrb in contrast is significantly bigger. Therefore, the reactions are stronger spatially correlated upon lowering the temperature. Table 1. The diffusion constant, rate of reactions ξ , C4, C5 are the fraction (%) of four- and five-told coordination, respectively; Cother is fraction of another type coordination (3 or 6). DSi , cm2/s 1.70×10-7 2.79×10-6 Temperature, K 2600 3500 DO , cm2/s 2.32×10-7 3.95×10-6 ξ 4.07×10-4 1.03×10-3 C4 93.90 86.37 C5 5.87 12.70 Cother 0.23 0.93 Table 2. The main characteristics of coordination cells of UNBRB groups at 3500 K and tDH = 5×104 steps; Here Ngroup is the number of coordination cells; is the mean square displacement of CC atom; is the averaged number of reactions happening in the coordination cell. Group Ngroup 389 187 80 9 1 0 1 2 3 4 SiOx ,Å2 5.68 3.80 3.22 4.55 0.08 96.25 60.07 37.69 16.78 12.00 Ngroup 994 298 40 - OSiy ,Å2 7.69 4.04 2.22 - 45.87 20.46 5.40 - We have calculated the number of unbroken bonds (UNBRB) in each coordination cell for time tDH. The coordination cells can be divided into UNBRB groups. The kth group includes coordination cells that have k unbroken bonds. The main characteristics of UNBRB groups for the case tDH = 5×104 steps (23.9 ps) are presented in Table 2. The atoms in the system move in average on the distance of 1.52σ; here σ is the position of first peak of PRDF for pair Si-O; σ = 1.62 Å. As shown in Table 2, there are a large number of CC atoms belonging to kth groups with k > 0. During time tDH these CC atoms and its neighbors move like a "large molecule" which consists of CC atom and k neighboring atoms. The liquid likely contains different types of "molecule" such as Si, SiOx, O and OSiy where x=1, 2, 3, 4; y=1, 2. The parameter k plays the role of the size of "molecules". It is interesting to note that CC atoms in the group with smaller parameter k move in average over larger distance (larger ). Moreover, the larger , the smaller parameter k is. It means that the number of reactions happening in the coordination cell is strongly correlated with the mobility of the CC atom. The "small molecule" is more mobile than "large molecule". The existence of different size "molecules" within long time tDH is related to the DH in the liquid. 2000 a b 5000 1500 mcc 4000 2600 K 3500 K 1000 3500 K, mresb+ munbrb 3000 3500 K, munbrb 2600 K, mresb+ munbrb 2000 500 2600 K, munbrb 1000 0 0 20000 40000 60000 80000 100000 The number of reactions 0 20000 40000 60000 80000 100000 The number of reactions Fig. 3. The dependence of mcc (a) and the dependence of munbrb and munbrb+mresb (b) as a function of number of reactions. 36 N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 The number of rigid clusters 160 120 4 4 5x10 steps 2x10 steps 80 40 0 0 2 4 6 8 0 50 100 150 200 The number of atoms in a cluster 150 100 50 The mean square 2 displacement, Å The number of clusters Fig.4. The size distribution of rigid clusters for the cases of tDH = 2×104 and 5×104 steps. The temperature is 3500 K. 0 A2 6 A1 3 0 0 10000 20000 30000 40000 50000 Steps Fig.5. The number of clusters MCL and mean square displacement per atoms at 11 moments. The temperature is 3500 K. If a number of "molecules" link to each other by unbroken bonds, then it forms large rigid cluster. Fast atoms move through the paths where no large rigid clusters locate; slow atoms locate near or inside large rigid clusters. Because large rigid clusters spontaneously decay with times, hence the size distribution of rigid clusters depends on the time tDH. In Fig.4, we show the size distribution of rigid clusters. The curves for different time tDH significantly differ from each other. In particular, for tDH = 5×104, the largest cluster found has the size of 6 atoms, meanwhile in the case of tDH = 2×104 steps, the size of largest cluster is 181 atoms. For tDH = 5×104 we consider a set A2 of atoms which belong to the rigid clusters composed of 4, 5 and 6 atoms. The total number of atoms in A2 is 156. We also consider the set A1. It consists of 156 atoms which are randomly chosen from the system. The clusters for A1 and A2 are determined at 11 N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 37 The number of atoms in largest cluster The number of clusters The mean square diplacement, Å2 moments. The result is shown in Fig.5. In the case of A1, the number of clusters found varies from 133 to 137. The largest cluster has the size of 4 atoms. Meanwhile for A2, the number of clusters varies from 24 to 27 that significantly smaller than of A1. The largest cluster found has 16 atoms. It follows that large rigid clusters do not move in different places in the system, but tend to locate together in some small volume. On the other hand, the mean square displacement per atom for A 2 is significantly smaller than for A1 (see Fig.5). It means that the groups of immobile atoms locate in some small volumes for 5×104 steps. This indicates the existence of DH. The reason of DH observed therefore is the rigid clusters which are formed due to none-uniform distribution of reactions in the space. In the case of 2600 K and tDH = 2×105 steps (95.6 ps) we found rigid clusters composing of 268 and 216 atoms (about 24% all atoms in the system). It follows that as the temperature decreases rigid clusters in the liquid are bigger and exist for longer times. If one continues cool down the liquid, this rigid cluster becomes percolate over whole system. Because the atoms of large rigid clusters are immobile, hence the dynamics in the liquid decreases dramatically. On the other hand, due to small change in the structure the rate of reactions decreases much weaker. So we can suggest that the dynamical slowing down may be related to the percolation rigid cluster present near glass-transition 20 a 15 10 5 0 240 0 10000 20000 30000 40000 50000 180 120 60 b A1 A3 A4 0 24 0 10000 20000 30000 40000 50000 10000 20000 30000 40000 50000 16 8 c 0 0 Steps Fig.6. The mean square displacement per atom, the number of clusters MCL and the number of atoms in largest cluster MACL for A1, A3 and A4. N.T.T. Ha et al. / VNU Journal of Science: Mathematics – Physics, Vol. 32, No. 4 (2016) 32-39 38 We consider three sets A1, A3, A4 which have 291 atoms (about 14% all atoms in the system). The A3, A4 include the most mobile and immobile atoms, respectively. We have calculated MCL, MACL and mean square displacement per atom at 11 moments during 5×104 steps. The result is presented in Fig.6. As shown in Fig.6a, the mean square displacement per atom for A3, A4 significantly differs from ones of A1. It means that A3, A4 represent the set of mobile or immobile atoms. The MCL for A1 varies from 225 to 240 that significantly larger than ones for A3 and A4. The MACL of A1 in contrast is significantly smaller. As shown in Fig.6b and 6c, MACL for A3 and A4 varies less 20 atoms. MCL for A3 and A4 varies from 8 to 24. It is interesting to clarify how large clusters behave in the space with times. For A3 the cluster may split or merge during time tDH. Therefore, the motion of atoms of A3, A4 evidences the DH in the liquid. 4. Conclusions The MD simulation shows that the diffusion is realized via reactions SiOxSiOx1 and OSiy OSi x1. As the temperature decreases, the diffusion constant decreases much faster than the rate of reactions. The reason is that the reactions happen not randomly in the system, but they occur more frequently on some atoms and bonds Si-O. Moreover, a large amount of reactions occur to restore the bonds and the diffusion constant depends on the fraction of those reactions. The coordination cells can be divided into UNBRB groups. The CC atoms in the group with smaller parameter k move in average over larger distance. Here k is the number of unbroken or restored bonds in the coordination cell. It was shown that the number of reactions happening in the coordination cell is strongly correlated with the mobility of CC atoms. We found rigid clusters which exist in the liquid for long time tDH. The atoms of large rigid clusters are immobile. Until the times tDH when the number of rigid bonds is large, there are specific microscopic regions in the liquid. The network structure of these regions changes a little and the atoms in the region move slowly. The clusters of slow atoms are observed there. The atom rearranging occurs outside those regions. The simulation shows that the DH can be seen from the sets of: (1) most mobile or immobile atoms; (2) atoms belong to rigid clusters. The effect is most pronounced from the atoms of rigid clusters. The DH can be seen by atoms of rigid clusters until the sum of rigid and restored bonds is enough small. Acknowledgements The authors are grateful for support by Hanoi University of Science and Technology. 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