A novel difference between strong liquids and fragile liquids in their dynamics near the glass transition

Physica A 442 (2016) 1–13

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Physica A journal homepage: www.elsevier.com/locate/physa

A novel difference between strong liquids and fragile liquids in their dynamics near the glass transition Michio Tokuyama a,∗ , Shohei Enda b , Junichi Kawamura a a

Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan

b

13-36-18, Towada City, Aomori, Japan

highlights • The dynamics of fragile liquids and that of strong liquids is compared consistently. • A novel difference between their dynamics is found near glass transition. • Existence of master curve in each type of liquids is proposed.

article

info

Article history: Received 12 May 2014 Received in revised form 11 June 2015 Available online 8 September 2015 Keywords: Diffusion coefficient Fragile liquids Glass transition Master curve Strong liquids Universality

abstract The systematic method to explore how the dynamics of strong liquids (S) is different from that of fragile liquids (F) near the glass transition is proposed from a unified point of view discussed recently by Tokuyama. The extensive molecular-dynamics simulations are performed on different glass-forming materials. The simulation results for the mean-nth displacement Mn (t ) are then analyzed from the unified point of view, where n is an even (α) number. Thus, it is first shown that in each type of liquids there exists a master curve Hn n (α) as Mn (t ) = R Hn (vth t /R; D/Rvth ) onto which any simulation results collapse at the same value of D/Rvth , where R is a characteristic length such as an interatomic distance, D a longtime self-diffusion coefficient, vth a thermal velocity, and α = F and S. The master curves (S ) (F ) Hn and Hn are then shown not to coincide with each other in the so-called cage region even at the same value of D/Rvth . Thus, it is emphasized that the dynamics of strong liquids is quite different from that of fragile liquids. © 2015 Elsevier B.V. All rights reserved.

1. Introduction It has been known for a long time since Angell [1] has proposed a famous classification in viscosities of glass-forming materials that there exist two types of glass-forming liquids, fragile liquids (F) and strong liquids (S), near the glass transition [2–12]. The systems with short-range interactions such as o-terphenyl and glycerol are typical examples of fragile liquids, while the covalently bonded network glass formers such as SiO2 and GeO2 are known as typical examples of strong liquids. Thus, it has been understood commonsensically since then that the transport coefficients of both liquids, such as viscosity and self-diffusion coefficient, are well described by the Vogel–Fulcher–Tammann (VFT) law [13–15], although the fitting temperature range for strong liquids is shorter than that for fragile liquids. However, it is not clear yet how the dynamics of strong liquids is different from that of fragile liquids in a supercooled state. Thus, it is still important to clarify it not only qualitatively but also quantitatively from a unified point of view.

∗

Corresponding author. E-mail address: [email protected] (M. Tokuyama).

http://dx.doi.org/10.1016/j.physa.2015.08.046 0378-4371/© 2015 Elsevier B.V. All rights reserved.

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M. Tokuyama et al. / Physica A 442 (2016) 1–13

In order to classify the long-time self-diffusion coefficient D(T ) into two types of glass forming liquids from a unified point of view consistently, Tokuyama [16–18] has recently shown that the α - and β -relaxation times, τα and τβ , obey power laws in a supercooled state

τα ∼ D−(1+µ) ,

τβ ∼ D−(1−µ) ,

(1)

where the exponent µ is given by µ ≃ 1/5 for (F) and 2/11 for (S). Then, the following master curve f (x; η) for D(T ) has been proposed: D(T ) = d0 f (Tf /T ; η), f (x; η) =

( 1 − x)

2+η

x

(2) exp[62x3+η (1 − x)2+η ],

(3)

where Tf is a fictive singular temperature to be determined and d0 a positive constant to be determined. Here the exponent η is given by η = 2(1 − 3µ)/3µ, which leads to η ≃ 4/3 for (F) and 5/3 for (S). Thus, it has been shown by analyzing many different data that both types of liquids are well described by two types of master curves up to the deviation point Tn , below which all the data start to deviate from them and obey the Arrhenius law, where Tn > Tf . Here we note that Tn coincides with the so-called thermodynamic glass transition temperature Tg and the master curves can be also fitted by the VFT law well for T ≥ Tn [17,18]. Thus, all the diffusion data in each type collapse onto each single master curve f (x; η) (see Fig. 1). Their material differences are just characterized by a set of parameters (Tf , d0 , η). From this viewpoint, therefore, those parameters may correspond to the so-called degree of fragility usually discussed among different systems [2,6,19–21]. Here we should mention that the present approach gives a mathematical tool to distinguish two types of liquids from each other and the classified results agree with those obtained by Angell. In the present paper we only discuss the dynamics of two different types of glass-forming materials, fragile glass formers (F) and network strong glass formers (S), from a unified point of view [22]. In order to compare the dynamics of F with that of S, it is convenient to use the mean-nth displacement Mn (t ) given by Mn (t ) = ⟨|Xiα (t ) − Xiα (0)|n ⟩, where Xiα (t ) is a position vector of ith atom α at time t, the brackets an average over the equilibrium ensemble, and n even numbers. Analyses of (α) many data then suggest an existence of a master curve Hn for Mn (t ) in each type as Mn (t ) = Rn Hn(α) (vth t /R; D/Rvth ),

(4)

where R is the characteristic length such as an interatomic distance, vth the average thermal velocity, and α = F and S. Any (α) data in each type are thus shown to collapse onto a single master curve Hn at the same value of D/Rvth . Then, we also show (β) (α) that even at the same value of D/Rvth the master curve Hn for type α does not coincide at all with Hn for other type β(̸=α) in the cage region for τf ≤ t ≤ τβ , in which each particle behaves as if it is trapped in a cage mostly formed by neighboring (α)

particles, where τf is a mean-free time before which each particle undergoes a ballistic motion. On the other hand, Hn

and

(β)

Hn (α ̸= β) are easily shown to coincide with each other both for a short-time region (t ≪ τf ) and for a long-time region (τβ ≪ t) at the same value of D/Rvth . In fact, for both time regions we have Hn(α) (τ ) =

(n + 1)!(3!)−n/2 (α) n/2 H2 (τ ) (n/2)!

(5)

with (α)

H2 (τ ) ≃



3τ 2 , 6(D/Rvth )τ

(t ≪ τf ) (t ≫ τβ ),

(6)

where τ = vth t /R. Thus, we emphasize that an explicit disagreement between different types appears only in the cage region. Finally, we note that although the even number n is taken up to 6 here for simplicity, the same results as those discussed in the present paper also hold for n ≥ 8. We begin in Section 2 by briefly reviewing the mean-field theory recently proposed. We first discuss the mean-field equation for the mean-square displacement and its related characteristic times. Then, we show two types of master curves for the long-time self-diffusion coefficient. One is a master curve for fragile liquids and another is for strong liquids. In Section 3, we introduce several potentials to perform extensive molecular-dynamics simulations. In Section 4, we briefly review how physical quantities satisfy the universality near the glass transition. Based on such a universality, we then show (i) that there exist a master curve Hn for the mean-nth displacement in each liquid, (F) and (S). In Section 5, we show that the (β) (α) master curves Hn and Hn in different types α and β(̸=α) do not coincide with each other in the cage region even at the same value of D/Rvth . We conclude in Section 6 with a summary. 2. Mean-field theory Here we briefly summarize the mean-field theory of the glass transition (MFT) for molecular systems recently proposed by Tokuyama [16,17,22–25]. The mean-field theory consists of the following two essential points: (A) Mean-field equation

M. Tokuyama et al. / Physica A 442 (2016) 1–13

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Fig. 1. (Color online) A log–log plot of D/d0 versus scaled temperature Tf /T for fragile liquids (F) and strong liquids (S). The symbols indicate the simulation results; for (F) () A in A80 B20 (LJ), (+) Al in Al2 O3 , (×) O in Al2 O3 , (•) A in A80 B20 (SW), and (◦) B in A80 B20 (SW), and for (S) () Si in SiO2 (BKS), (+) O in SiO2 (BKS), (×) Si in SiO2 (NV), and (◦) O in SiO2 (NV). The solid line indicates the master curve f (x) given by Eq. (19). The relevant parameters Tf and d0 are listed in Table 4.

for M2 (t ) and (B) Two different types of singular functions for D(T ), the mean-field curve g (Tl /T ) for a liquid state and the master curve f (Tf /T , η) for a supercooled state, where Tl and Tf are fictive singular temperatures to be determined and Tl > Tf . 2.1. Mean-field equation The mean-square displacement M2 (t ) of ith particle A in molecular systems {ABC · · ·} is described by a nonlinear equation [22] d dt

2 M2 (t ) = 6D + 6[vth t − D]e−M2 (t )/ℓ , 2

(7)

where ℓ is a mean-free path of particle A over which the particle can move freely by a ballistic motion and vth (=(kB T /m)1/2 ) the average thermal velocity. Eq. (7) can be easily solved to give a formal solution

 M2 (t ) = 6Dt + ℓ ln e 2

−6t /τβ

+

1 6



τβ τf

2  1−

 1+

6t

τβ

 e

−6t /τβ



,

(8)

where τβ (=ℓ2 /D) denotes a time for a particle to diffuse over a distance of order ℓ with the diffusion coefficient D and is identical to the so-called β -relaxation time. Here τf (=ℓ/vth ) is a mean-free time, within which each particle can move freely by a ballistic motion. The solution (8) satisfies the asymptotic forms given by Eq. (6). As shown in Ref. [24], the meanfree path ℓ is uniquely determined by D/(Rvth ). Hence the solution (8) suggests that the dynamics is described by only one

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M. Tokuyama et al. / Physica A 442 (2016) 1–13 Table 1 Universal value uˆ i . Type

uˆ β

uˆ g

Fragile Strong

2.833 2.693

5.0 4.0

parameter D/(Rvth ) if the length and the time are scaled by R and τth (=R/vth ), respectively. This means that the dynamics in different systems coincides with each other if D/(Rvth ) has the same value in them. Hence this is called a universality in dynamics. Here we should note that this universality holds even in a supercooled state, although it is predicted using the mean-field equation (7) which is valid only for a liquid state. Since the single-particle dynamics is determined by only one parameter D/(Rvth ), it is convenient to introduce a new parameter uˆ by [25] uˆ = log10 (Rvth /D).

(9)

As uˆ increases (or T decreases), there exist three states; a liquid state [L] for uˆ < uˆ β (or Ts < T ), a supercooled state [S] for uˆ β ≤ uˆ < uˆ g (or Tg < T ≤ Ts ), and a glass state [G] for uˆ g ≤ uˆ (or T ≤ Tg ), where Ts is a supercooled point and Tg a glass transition point. The values of uˆ i are listed in Table 1. Here uˆ β (or Ts ) is determined by the intersection point of the mean-field curve g (Tl /T ) with the master curve f (Tf /T ) [25] and coincides with a peak position of a specific heat, while uˆ g (or Tg ) is determined by a deviation point Tn at which the simulation results and the experimental data for the long-time self-diffusion coefficient start to deviate from the master curve f (Tf /T ). Here Tn coincides with the thermodynamic glass transition point [18]. Thus, the mean-field fitting value for the β -relaxation time τβ /τth is uniquely determined by uˆ . In general, however, the length R is not known. As a well-known example in which R is known, one can take the Lennard-Jones (LJ) binary mixtures A80 B20 , where the LJ potential Uαβ (r ) is given by Uαβ (r ) = 4εαβ [(σαβ /r )12 − (σαβ /r )6 ].

(10)

Here σAA = σ , εAA = ε, σAB = 0.8σ , εAB = 1.5ε, σBB = 0.88σ , and εBB = 0.5ε , where σ is a length unit and ε an energy unit [26]. Then, one can choose σ as R for A particle. Thus, one can use the simulation results for the LJ binary mixtures as reference to determine R for fragile systems based on the universality. This will be discussed later. 2.2. Master curve for long-time self-diffusion coefficient In this subsection, we briefly review two types of master curves for D. In order to distinguish the strong liquids from the fragile liquids consistently, Tokuyama has recently analyzed the structural relaxation time τα and the β -relaxation time τβ for self-diffusion in different glass-forming liquids and has proposed two types of master curves for the self-diffusion near the glass transition [16–18]. Here τα is defined as a time on which the self-intermediate scattering function FS (q, t ) decays to e−1 of its initial value, that is, FS (q, τα ) = e−1 , while τβ is a time on which the particles can escape from their cages [25]. In a liquid state [L], the relaxation times τα and τβ are then shown to obey power laws

τα ∼ τβ ∼ D−(1−ν) , (11) where the exponent ν is obtained by fitting as ν ≃ 1/3. On the other hand, in [L] the experimental data and the simulation results can be well described by the mean-field singular function [25,27] D(T ) ∝ g (Tl /T ) ∝

  T

Tl

1−

Tl T

2

∼ ϵ02 ,

(12)

where Tl is a singular temperature to be determined by fitting and ϵ0 = 1 − Tl /T . In a supercooled state [S], the relaxation times τα and τβ are shown to obey power laws

τα ∼ D−(1+µ) , τβ ∼ D−(1−µ) , (13) where the exponent µ is obtained by fitting as µ ≃ 1/5 for fragile liquids and 2/11 for strong liquids. We now assume that as long as the system is in equilibrium, the long-time self-diffusion coefficient D(T ) obeys the following singular function in [S]:   2+η Tf T D(T ) ∝ 1− ∼ ϵ 2+η , (14) Tf

T

where Tf (
ℓ ∼ ϵ0ν in [L], ℓ∼ϵ

(2+η)µ/2

(15) in [S].

(16)

M. Tokuyama et al. / Physica A 442 (2016) 1–13

5

The detailed analyses [27,24] show that ℓ obeys the same power law in both states because the caging mechanism is not changed in both states. Thus, use of Eqs. (15) and (16) leads to

ν = (2 + η)µ/2 or η = 2(ν/µ − 1).

(17)

Then, one finds η ≃ 4/3 for fragile liquids and 5/3 for strong liquids. Thus, Eq. (14) can describe the self-diffusion data in a supercooled state, while Eq. (12) holds in a liquid state. The intersection point of Eq. (14) with Eq. (12) thus determines a supercooled point Ts (or uβ ). In order to find an asymptotic function which holds in both states, we assume that D(T ) can be written as D(T ) = d0 f (Tf /T ; η),

(18)

where d0 is a positive constant to be determined. Then, the function f (x) must numerically coincide with Eq. (12) in [L] and Eq. (14) in [S]. As shown in Ref. [17], expanding f (x) in powers of ϵ 2+η /x, one can thus find the master curve f (x; η) ≃

(1 − x)2+η

exp[62x3+η (1 − x)2+η ], (19) x where x = Tf /T . We note here that the power-law exponent for strong liquids is slightly different from that for fragile liquids. Although the quantitative difference between exponents in both liquids is small, it is important to show that there exist qualitatively different mechanisms between them since the exponents should result from the many-body correlations due to the nonlinear density fluctuations. In the previous paper [17], we have shown that Eq. (19) can describe any data for self-diffusion coefficient in fragile and strong liquids, up to the deviation temperature Tn , below which the system becomes out of equilibrium and all the data start to deviate from the master curve. If the data are scaled by d0 and Tf , therefore, they are all collapsed onto two types of master curves given by Eq. (19), a fragile master curve with η = 4/3 and a strong master curve with η = 5/3 (see Fig. 1). Hence one can classify the microscopic differences among various liquids by a set of parameters (Tf , d0 , η). Thus, Eq. (19) may give a mathematical expression of Angell’s classification. Finally, we should note that the present situation is quite different from that obtained by the VFT law. In fact, that law can describe the data not only for fragile liquids but also for strong liquids by adjusting the three fitting parameters. Since one cannot eliminate those fitting parameters appropriately, however, one cannot obtain a single master curve such as that shown in Fig. 1, where T is scaled by Tf and D by d0 . In the previous papers [16–18], we have investigated many different glass-forming materials from a unified point of view based on two types of master curves f (x; η) and classified them into two types of liquids, fragile liquids with η = 4/3 and strong liquids with η = 5/3. Thus, one can express Angell’s classification from a different point of view. However, this classification has a weak point in which the difference between their exponents η is very small to distinguish two types of liquids quantitatively. Hence very precise analyses are required to make it even under the situation that the experimental data and the simulation results always have fluctuations. In the following, therefore, we first perform the extensive molecular-dynamics simulations on different glass-forming materials and then investigate their dynamics fully by calculating the mean-nth displacement Mn (t ). Thus, we show how the dynamics of strong liquids is different from that of fragile liquids consistently from a unified point of view based on the universality. 3. Molecular-dynamics simulations In order to investigate the differences between fragile liquids and strong liquids, we perform the extensive moleculardynamics simulations under the so-called NVT method with periodic boundary conditions on the following different systems: For fragile liquids we take binary mixtures A80 B20 with the Stillinger–Weber (SW) potential [28] and Al2 O3 with the Born–Meyer (BM) potential [29]. We also use the previous simulation results on the LJ binary mixtures [30,31] as reference. On the other hand, for strong liquids we take SiO2 with the Beest–Kremer–Santen (BKS) potential [32] and also SiO2 with the Nakano–Vashishta (NV) potential [33] as a typical example of network glass formers. The SW potential is given by

   −1    r r σαβ 12   − 1 exp − rc for < rc , εαβ r σαβ σαβ (20) Uαβ (r ) =  r   > rc , 0 for σαβ where α, β ∈ {A, B}. Here the parameters εαβ , σαβ , and rc are given by σAA = σ , εAA = 8.8ε, σAB = 0.8σ , εAB = 13.2ε, σBB = 0.88σ , εBB = 4.4ε , and rc = 1.652. Here ε is an energy unit and σ a length unit. The system contains N = 10 976 particles, which is composed of NA = 8780 particles of type A with mass mA and NB = 2196 particles of type B with mass mB . Length, time, and temperature are scaled by σ , t0 (=σ /v0 ), and ε/kB , respectively, where v0 = (ε/mA )1/2 . The simulations are performed in a cubic box of length 20.89σ with periodic boundary conditions, where the number density is 1.2. The BKS and the BM potentials are given by Uαβ (r ) =

qα qβ rαβ

+ Aαβ exp(−bαβ rαβ ) −

cαβ 6 rαβ

,

(21)

6

M. Tokuyama et al. / Physica A 442 (2016) 1–13 Table 2 Potential parameters for BKS [32] and BM [29]. −1

qα (e)

Aαβ (eV)

bαβ (Å

BKS

Si–Si O–Si O–O

2.4 – −1.2

0.0000 18 003.7572 1388.7730

0.00000 4.87318 2.76000

BM

Al–Al Al–O O–O

3 – −2

0.00 1779.86 1500.00

3.448 3.448 3.448

)

6

cαβ (eV Å ) 0.0000 133.5381 175.0000 0.0000 0.0000 0.0000

Table 3 Potential parameters for NV [33].

Si–Si O–Si O–O O–Si–O Si–O–Si

ϵ (eV)

aαα (Å)

Zα (e)

aα (Å3 )

nαβ

Bα (eV)

θα

1.592 1.592 1.592 – –

0.47 – 1.2 – –

−0.88

0.00 – 2.4 – –

11 9 7 – –

– – – 4.993 19.972

– – – 109.47 141.00

– 1.76 – –

where the potential parameters are listed in Table 2. For the BKS potential, the system contains N = 3000 particles in the cubic box of volume L3 , which is composed of NSi = 1000 particles of Si with mass mSi = 4.66 × 10−26 (kg) and NO = 2000 particles of O with mO = 2.66 × 10−26 (kg), where L = 34.79 Å. For the BM potential, the system contains N = 3000 particles in the cubic box of volume L3 , which is composed of NAl = 1200 particles of Al with mass mAl = 4.5 × 10−26 (kg) and NO = 1800 particles of O with mO = 2.66 × 10−26 (kg), where L = 32.02 Å. Those system sizes are enough to avoid a finite size effect in strong liquids [34]. The NV potential is given by U =

 α<β

(2)

Uαβ +

(3)

 α,β<γ

Uαβγ

(22)

with the two-body potential (2) Uαβ (r )

=ϵ



aαα + aββ r

nαβ +

aα Zβ2 + aβ Zα2 −r /A Zα Zβ −r /A0 1 e − e , r 2r 4

(23)

and the three-body potential



(3)

Uαβγ = Bα exp

1 rαβ − A2

+

1 rαγ − A2



rαβ · rαγ − cos θ α rαβ rαγ

2

θ (A2 − rαβ )θ (A2 − rαγ ),

(24)

where θ (x) is a step function, A0 = 4.43 (Å), A1 = 2.5 (Å), A2 = 5.5 (Å). The potential parameters are listed in Table 3. The system contains N = 5184 particles in the cubic box of volume L3 , which is composed of NSi = 1728 particles of Si with mass mSi = 4.66 × 10−26 (kg) and NO = 3456 particles of O with mO = 2.66 × 10−26 (kg), where L = 42.996 Å. The Newton equations are solved by the velocity Verlet algorithm under the NVT ensemble for each system. The simulations are repeated until the system is equilibrated, where the time scale of equilibration is of order 10 ns for BKS, BM, and NV, and of order 105 t0 for SW. Next we analyze those simulation results from a unified point of view based on the universality. 4. Universality near the glass transition We now analyze the simulation results in two different types of liquids, fragile liquids and strong liquids, from a unified point of view based on the universality and then show how the dynamics of strong liquids is different from that of fragile liquids. In order to discuss the differences between strong liquids and fragile liquids, we investigate the following physical quantities. The first is the mean-nth displacement given by Mn (t ) = ⟨|Xiα (t ) − Xiα (0)|n ⟩,

(25) α

where the brackets indicate the average over an equilibrium ensemble, Xi (t ) the position vector of ith particle α at time t, and n even numbers. The second is the long-time self-diffusion coefficient D(T ) given by D(T ) = lim

t →∞

M2 (t ) 6t

.

(26)

M. Tokuyama et al. / Physica A 442 (2016) 1–13

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Table 4 Characteristic length R, singular temperature Tf , and d0 for different systems. Type

System

R

Tf

d0

Fragile (η = 4/3)

A (LJ) Al (BM) O (BM) A (SW) B (SW)

1.000 2.928 (Å) 3.047 (Å) 1.000 1.250

0.3955 1955 (K) 1916 (K) 0.5153 0.4886

0.0283 0.8394 × 10−8 (m2 /s) 1.3405 × 10−8 (m2 /s) 0.0267 0.0357

Strong (η = 5/3)

Si (NV) O (NV) Si (BKS) O (BKS)

3.717 (Å) 4.506 (Å) 3.840 (Å) 4.650 (Å)

2612 (K) 2621 (K) 2904 (K) 2865 (K)

4.4367 × 10−8 5.7416 × 10−8 5.4967 × 10−8 6.5068 × 10−8

(m2 /s) (m2 /s) (m2 /s) (m2 /s)

The third is the static structure factor Sαβ (q) given by Sαβ (q) =

1  N i=1 j= ̸ i

β

⟨exp[iq · {Xiα (0) − Xj (0)}]⟩.

(27)

Uing (26), one can first obtain the temperature dependence of D(T ) for each system. In Fig. 1, all the simulation results for D(T ) are shown to collapse onto two types of master curves given by Eq. (19), where Tf and d0 are listed in Table 4. As discussed in the previous paper [18], the deviation point Tn at which all data start to deviate from the master curve coincides with the thermodynamic glass transition point Tg . For fragile liquids it is given by Tg ≃ Tf /0.938 at f (Tf /Tg ) ≃ 10−4.0 , while Tg ≃ Tf /0.855 at f (Tf /Tg ) ≃ 10−3.0 for strong liquids. Using Eq. (8), one can next obtain the mean-field fitting values for the β -relaxation time τβ . From a unified point of view based on the universality discussed in the previous paper [25], the dimensionless time τβ /τth for different systems should coincide with each other at the same value of uˆ . Since the characteristic length R is not known, however, the time τth is not known yet. In order to find R in fragile liquids, as reference one can use the dimensionless time for A particle obtained by the simulations on the SW binary mixtures [35] or the LJ binary mixtures [30,31] since R is known as R = σ . In fact, it satisfies the power laws given by Eqs. (11) and (13), which are described by the straight lines given in Fig. 2(F). Then, the value of the characteristic length R for an arbitrary particle is chosen for the dimensionless time τβ /τth to obey those power-law lines as a function of uˆ . In strong liquids one may take R ≃ 3.717 (Å) for Si (NV) as reference just for convenience. In fact, its dimensionless time satisfies the power laws given by Eqs. (11) and (13), which are described by the straight lines given in Fig. 2(S). Similarly to F, R for other particles is then chosen for the dimensionless time τβ /τth to obey those power-law lines as a function of uˆ . In Fig. 2, all the data for the dimensionless time τβ /τth are then shown versus uˆ in each type of liquids, (F) and (S), where the fitting value of R for each atom is listed in Table 4. Thus, there exists a small difference between fragile liquids and strong liquids. This difference would be roughly explained to result from the fact that the network strong liquids such as SiO2 have an open tetrahedral network, while the fragile liquids do not [36–39,8,40–43]. The static properties of SiO2 are known to be reproduced by employing the NV potential [33] and also the BKS potential [44–53]. As is shown in Fig. 3, the static structure factor Sαα (q) in network strong liquids usually has the so-called first sharp diffraction peak [54], while it does not in fragile liquids. In the following, we discuss two types of liquids, (F) fragile liquids and (S) strong liquids, separately, and then show that in each type of liquids any data coincide with each other at the same value of uˆ [16,17]. Thus, the detailed analyses suggest (α) an existence of a master curve Hn for the mean-nth displacement Mn (t ) given by Mn (t ) = Rn Hn(α) (t /τth ; uˆ ),

(28)

where α = F and S. Thus, all the simulation data seem to collapse onto each single master curve at the same value of uˆ , although its analytic form is not known. 4.1. Master curve in fragile liquids We first show the master curve in fragile liquids. In Fig. 4, the simulation results for the scaled mean-nth displacement Mn (t )/Rn are plotted versus t /τth for different values of uˆ in fragile liquids, Al2 O3 , A80 B20 (LJ), and A80 B20 (SW), where n = 2, 4, and 6. At each value of uˆ , the simulation results in different systems coincide with each other within error. Thus, it is (F ) (F ) suggested that there exists a master curve Hn (t /τth ; uˆ ) for any fragile liquids, where the existence of H2 for many different fragile systems has already been discussed in the previous papers [16,17]. 4.2. Master curve in strong liquids We next show the master curve in strong liquids. In Fig. 5, the simulation results for the mean-nth displacement Mn (t )/Rn are plotted versus t /τth for different values of uˆ in network glass formers, SiO2 (BKS) and SiO2 (NV), where n = 2, 4, and 6.

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M. Tokuyama et al. / Physica A 442 (2016) 1–13

Fig. 2. (Color online) A log–log plot of τβ /τth versus uˆ for fragile liquids (F) and strong liquids (S). The symbols indicate the simulation results; for (F) () Al in Al2 O3 , (◦) O in Al2 O3 , (+) A in A80 B20 (LJ), (⊙) A in A80 B20 (SW), and () B in A80 B20 (SW), and for (S) () Si in SiO2 (NV), (◦) O in SiO2 (NV), (△) Si in SiO2 (BKS), and () O in SiO2 (BKS). The solid lines indicate straight lines y = 0.8x − 1.45 for (F) and y = (9/11)x − 1.55 for (S), and the dotted lines y = (2/3)x − 1.091 for (F) and y = (2/3)x − 1.142 for (S).

Fig. 3. (Color online) A plot of SOO (q) and SAA (q) versus scaled wave vector qR around uˆ ≃ 3.143 for fragile liquids and strong liquids. The solid line indicates SOO (q) for SiO2 (NV) at T = 3500 (K), the dashed line SOO (q) for SiO2 (BKS) at T = 3700 (K), the dot-dashed line SOO (q) for Al2 O3 at T = 2700 (K), and the long-dashed line SAA (q) for A80 B20 at T = 0.714.

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Table 5 Temperature versus uˆ in fragile liquids. State

uˆ

A (LJ)

Al (BM)

O (BM)

A (SW)

[L]

1.677 1.957 2.307 2.580

1.428 1.000 0.769 0.667

6000 (K) 4500 (K) 3600 (K) 3200 (K)

5500 (K) – 3400 (K) 3000 (K)

1.667 1.250 – 0.833

[S]

2.926 3.946

0.588 0.476

2800 (K) 2400 (K)

2700 (K) –

– 0.625

Table 6 Temperature versus uˆ in strong liquids. State

uˆ

Si (NV)

O (NV)

Si (BKS)

O (BKS)

[L]

1.793 1.913 2.085 2.220 2.474 2.693

5000 (K) – 4400 (K) 4200 (K) 3900 (K) 3700 (K)

– 5000 (K) – 4300 (K) – 3800 (K)

– 5200 (K) 4800 (K) 4600 (K) – –

5800 (K) – 5000 (K) – 4400 (K) –

[S]

2.955 3.021 3.284 3.322

3500 (K) – 3300 (K) –

3600 (K) – – –

– 3800 (K) 3600 (K) 3600 (K)

– 3800 (K) 3600 (K) 3600 (K)

Table 7 Temperature versus uˆ in fragile liquids and strong liquids. State

uˆ

Al (BM)

Si (NV)

Si (BKS)

[L]

1.779 1.842 1.969 2.302 2.369 2.494 2.638

5300 (K) 5000 (K) 4500 (K) 3600 (K) 3500 (K) 3300 (K) 3100 (K)

5000 (K) – – – – 3900 (K) 3700 (K)

– – – 4400 (K) – –

2.745 2.949 3.143 3.274

3000 (K) 2800 (K) 2700 (K) 2600 (K)

– 3500 (K) 3400 (K) 3300 (K)

4000 (K) – – –

[S]

At each value of uˆ , the simulation results in different systems coincide with each other within error. Thus, this suggests an (S ) existence of a single master curve Hn (t /τth ; uˆ ) for any strong glass formers. 5. Differences between fragile liquids and strong liquids (α)

In the previous section, we have shown that there exists a master curve Hn (t /τth ; uˆ ) in each type of liquids. In the present section, therefore, we compare the dynamics of strong liquids with that of fragile liquids at the same value of uˆ and explore how their dynamics is different from each other. In Fig. 6, the simulation results for the mean-nth displacement Mn (t )/Rn are plotted versus t /τth for different values of uˆ in fragile liquids Al2 O3 (BM) and network glass formers SiO2 (NV, BKS), where n = 2, 4, and 6. As a typical example, the dynamics of Si is compared with that of Al. In the β stage for τf ≤ t ≤ τβ , the dynamical behavior of Si is quite different from that of Al. This difference is caused by an open tetrahedral network in SiO2 . The results obtained here are also seen for other combinations between Al2 O3 and SiO2 . 6. Summary In the present paper, we have proposed the systematic method to investigate how the dynamics of strong liquids is different from that of fragile liquids. As examples of glass-forming materials, we have taken Al2 O3 , the LJ binary mixture A80 B20 , the SW binary mixture A80 B20 , and SiO2 (BKS and NV). We have first applied the mean-field theory for the simulation results in those different glass-forming materials. Then, we have obtained the long-time self-diffusion coefficient D(T ) and also the mean-field values of the characteristic time τβ . Using the master curve f (x; η), the diffusion coefficients in different systems have been safely classified into two types of liquids, fragile liquids and strong liquids. Thus, Al2 O3 , the LJ binary mixture A80 B20 , and the SW binary mixture A80 B20 were classified as fragile liquids with η = 4/3, while SiO2 (BKS and

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Fig. 4. (Color online) A log–log plot of scaled mean-nth displacement Mn (t )/Rn versus t /τth for different values of uˆ in fragile liquids. (a) n = 2 at uˆ ≃ 1.677, 1.957, 2.307, 2.580, 2.926, and 3.946, and (b) n = 4 and (c) n = 6 at uˆ ≃ 1.677, 1.957, 2.307, 2.580, and 2.926 (from top to bottom). The solid lines indicate the simulation results for Al (BM). The symbols indicate the simulation results for (•) A (LJ), () A (SW), and (+) O (BM). Temperature at each uˆ is listed in Table 5.

NV) was classified as strong liquids with η = 5/3 (see Fig. 1). In order to use the universality that all the dimensionless physical quantities must coincide with each other at the same value of the universal parameter uˆ (= log10 (Rvth /D)), we have then adjusted the unknown characteristic length R in each system so that the dimensionless time τβ /τth coincides with each

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Fig. 5. (Color online) A log–log plot of scaled mean-nth displacement Mn (t )/Rn versus t /τth for different values of uˆ in strong liquids. (a) n = 2 at uˆ ≃ 1.793, 2.085, 2.474, 2.693, 2.955, and 3.284, (b) n = 4 at uˆ ≃ 1.793, 2.085, 2.474, 2.693, 2.955, and 3.284, and (c) n = 6 at uˆ ≃ 1.793, 1.913, 2.085, 3.021, 3.322 (from top to bottom). The solid lines indicate the simulation results for Si (NV). The symbols indicate the simulation results for (•) O (NV), (+) O (BKS), and (◦) Si (BKS). Temperature at each uˆ is listed in Table 6.

other. The results are shown in Fig. 2 and Table 4. We have next investigated the simulation results for Mn (t ) in each type, (F) and (S), at the same value of uˆ . In type (F), all the simulation results have been shown to collapse onto a single master curve (F ) Hn at the same value of uˆ (see Fig. 4). In type (S) all the simulation results have been shown to collapse onto the master

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Fig. 6. (Color online) A log–log plot of scaled mean-nth displacement Mn (t )/Rn versus t /τth for different values of uˆ in fragile liquids Al2 O3 and strong liquids SiO2 . (a) n = 2 and (b) n = 4 at uˆ ≃ 1.779, 2.494, 2.638, 2.949, and 3.274, and (c) n = 6 at uˆ ≃ 1.669, 2.369, and 2.745 (from top to bottom). The solid lines indicate the simulation results for (a) Si (NV), (b) Si (NV), and (c) Si (BKS). The dotted lines indicate the simulation results for Al. The symbols indicate the relaxation times τf and τβ at uˆ = 3.274 for (a) and (b) and 2.745 for (c); () for Si and (◦) for Al. Temperature at each uˆ is listed in Table 7.

(S )

curve Hn at the same value of uˆ (see Fig. 5). Those classifications have been done based on the fact that the simulation results of type α do not coincide with those of different type β in the cage region even at the same value of uˆ (see Fig. 6). In fact, the disagreement between (F) and (S) is reasonable because their static structure factors have different structural properties from each other. Finally, from a unified point of view proposed in this paper one should also investigate the other

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interesting glass-forming materials such as Se which has a network structure but is usually believed to be a fragile liquid. This will be discussed elsewhere. Acknowledgments Authors (M.T. and J.K.) wish to thank Prof. Takashi Nakamura for his hospitality and encouragement. This work was partially supported by High Efficiency Rare Elements Extraction Technology Area, Institute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University, Japan. The simulations were performed using the SGI Altix3700Bx2 in Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

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