Noname manuscript No.(will be inserted by the editor)

Arvind Kumar Gautam, Nandlal Pingua, Aashish

Goyal, and Pankaj A. Apte

Dynamical instability causes the demise of a

supercooled tetrahedral liquid

July 21, 2017

SUPPLEMENTARY INFORMATION

Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, U.P, India 208016

2

-1.83

-1.79

-1.75

0 4 8 12

0.7

0.8

0.9

-1.83

-1.79

-1.75

0 4 8 12

0.7

0.8

0.9

-1.83

-1.79

-1.75

0 4 8 12

0.7

0.8

0.9

R SLR

[m, m]

[mid

]

b

f4b

f4cb

b

f4b

f4cb

[m, m]

[mid]

106 MC steps

Fig. S1 The NPT-MC trajectory at 205 K (with N = 4096 particles) and zero pressure. The block averages

b are over 2105 MC steps. Here is the per particle potential energy (in reduced units), and f4c = N4c/N

(please refer to the right ordinate) is the fraction of particles in the largest connected network of 4-coordinated

particles. The blue symbols denotes the points along the trajectory at which the local minimum [m, m] of

the probability distribution (corresponding to the liquid state) is accessed. This minimum is located within

the rectangular area formed by the points (m /2, m /2), and (m + /2, m + /2), where

m = 1.75465, m = 0.44946, = 4.4 104, and = 1.2 104. The point along the trajectory at

which straight line region is formed in the intermediate potential energy distribution (see Fig. S3) is denoted as

SLR. The red symbols denotes the configurations along the trajectory with the per particle potential energy

() in the interval [mid] = (mid /2, mid +/2). The vertical dashed line corresponds to the dynamical

crossover point along the trajectory as explained in the text.

3

0.445

0.447

0.449

0 4 8 12

0.13

0.17

0.21

0.445

0.447

0.449

0 4 8 12

0.13

0.17

0.21

R SLR

b

[m, m]

3B4c b

b

3B4c b

106 MC steps

Fig. S2 The block averages of the density , and the per particle 3-body energy of largest 4-coordinated

network 3B4c are shown for the same trajectory as in Fig. S1. The other symbols have the same meaning as in

Fig. S1.

4

0

4

8

12

-1.77 -1.76 -1.75 -1.74

10

12

-1.765 -1.76 -1.755

log

p(

)+co

nst

.

liquid

SLR

finalmid

mid

mid = 1.7614

R2 = 0.999999994

Tc = 203.537 K

Fig. S3 The potential energy distributions generated by the trajectory in Fig. S1. The blue stars represents

distribution upto R point (see Fig. S1). The intermediate distribution (denoted by pink square symbols)

containing the straight line regions (black squares) is the distribution upto the SLR point in Fig. S1. The x

(green) symbols represents final distribution obtained from trajectory upto 18.2 million MC steps. By final, we

mean that the distribution is not expected to evolve further since the system is unlikely to visit the part of the

phase space corresponding to the range of values in the above figure. The values of the correlation coefficient

R2 of the straight line fit, the mid point mid, and the configurational (or effective) temperature Tc of the first

SLR region (black squares) are given in the inset. For the second SLR region (at lower potential energy), the

values are : mid = 1.7680, R2 = 0.99994, and T c = 203.212 K.

5

0

4

8

12

-1.77 -1.76 -1.75 -1.74

8

10

12

-1.765 -1.76

log

p(

)+co

nst

.

mid

mid

Fig. S4 The figure shows the curves fitted to the SLR distribution of Fig. S3 according to the Taylor series

expansions [see Eq. (7) in the main text] around the various mid-points of the SLRs. The first order derivatives

a1 and a

1 are computed from the central difference technique and correspond to the values of the configurational

temperatures mentioned in the caption of Fig. S3. The second order derivatives a2 and a

2 are zero since the

expansion is around the mid-points of the SLRs. The values of the higher order derivatives are chosen so as to

achieve a reasonably good fit to the data and are as follows: (1) a3 = a4 = 0 and a5 = 4.0 107 for mid (2)

a3 = 0.00235 and a

4 = a

5 = 0 for

mid

6

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10.5 11 11.5 12 12.5

0.7

0.75

0.8

0.85

-1.81

-1.79

-1.77

-1.75

10.5 11 11.5 12 12.5

0.7

0.75

0.8

0.85

-1.81

-1.79

-1.77

-1.75

10.5 11 11.5 12 12.5

0.7

0.75

0.8

0.85

SLRR

[m, m]

[mid

]

b

f4b

f4cb

106 MC steps

Fig. S5 The zoomed in portion of the trajectory in Fig. S1. Here b denotes block averages over 62000

MC steps. The other symbols have the same meaning as in Fig. S1

7

0.446

0.448

0.45

10.5 11 11.5 12 12.5

0.12

0.16

0.2

0.24

0.446

0.448

0.45

10.5 11 11.5 12 12.5

0.12

0.16

0.2

0.24

R SLR

b 3B4c b

b

3B4c b

106 MC steps

Fig. S6 The zoomed in portion of the trajectory in Fig. S2. Here b denotes block averages over 62000

MC steps. The other symbols have the same meaning as in Fig. S2

8

-1.83

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-1.75

0 10 20 30 40 0.6

0.7

0.8

0.9

1

-1.83

-1.79

-1.75

0 10 20 30 40 0.6

0.7

0.8

0.9

1

-1.83

-1.79

-1.75

0 10 20 30 40 0.6

0.7

0.8

0.9

1R SLR

b

[m, m]

[mid]

f4b

f4cbb

f4b

f4cb

106 MC steps

Fig. S7 The NPT-MC trajectory at 205 K (with N = 1000 particles) and zero pressure. The block averages

b are over 2105 MC steps. Here is the per particle potential energy (in reduced units), and f4c = N4c/N

(please refer to the right ordinate) is the fraction of particles in the largest connected network of 4-coordinated

particles. The blue symbols denotes the points along the trajectory at which the local minimum [m, m] of

the probability distribution (corresponding to the liquid state) is accessed. This minimum is located within

the rectangular area formed by the points (m /2, m /2), and (m + /2, m + /2), where

m = 1.75421, m = 0.44958, = 4.4 104, and = 1.2 104. The point along the trajectory at

which straight line region is formed in the intermediate potential energy distribution (see Fig. S9) is denoted as

SLR. The red symbols denotes the configurations along the trajectory with the per particle potential energy

() in the interval [mid] = (mid /2, mid +/2). The vertical dashed line corresponds to the dynamical

crossover point along the trajectory as explained in the text.

9

0.443

0.447

0.451

0 10 20 30 40

0.1

0.14

0.18

0.22

0.443

0.447

0.451

0 10 20 30 40

0.1

0.14

0.18

0.22

R SLR

b[m, m]

3B4c

b

b

3B4c b

106 MC steps

Fig. S8 The block averages of the density , and the per particle 3-body energy of largest 4-coordinated

network 3B4c are shown for the same trajectory as in Fig. S7. The other symbols have the same meaning as in

Fig. S7.

10

0

4

8

12

-1.78 -1.76 -1.74

0

4

8

12

-1.78 -1.76 -1.74

13.1

13.3

13.5

-1.761 -1.759

13.1

13.3

13.5

-1.761 -1.759

log

p(

)+co

nst

.mid

mid

mid

mid

liquid(R)

SLR

final

Tc = 203.69K

mid = 1.7590

R2 = 0.99999998

Fig. S9 The potential energy distributions generated by the trajectory in Fig. S7. The blue stars represents

intermediate distribution upto R point (see Fig. S7). The intermediate distribution (denoted by pink square

symbols) containing the straight line region (black squares) is the distribution upto SLR point (see Fig. S7).

The x (green) symbols represent final distribution obtained from trajectory upto 42.4 million MC steps. By

final, we mean that the distribution is not expected to evolve further since after the system is unlikely to

visit the part of the phase space corresponding to the range of values in the above figure. The values of

the correlation coefficient R2 of the straight line fit, the mid point m, and the configurational (or effective)

temperature Tc of the first SLR region (black squares) are given in the inset. The corresponding values for

the SLR regions at lower potential energies are : mid = 1.7639, R2 = 0.999998, and T c = 202.788 K;

mid = 1.7683, R2 = 0.99997, and T c = 201.819 K;

mid = 1.7731, R2 = 0.99999, and T c = 200.341

K;

11

0

4

8

12

-1.78 -1.76 -1.74

0

4

8

12

-1.78 -1.76 -1.74

10

12

-1.772 -1.762 10

12

-1.772 -1.762

log

p(

)+co

nst

.midmidmidmid

Fig. S10 The figure shows the curves fitted to the SLR distribution of Fig. S9 according to the Taylor

series expansions [see Eq. (7) in the main text] around the various mid-points of the SLRs. The first order

derivatives a1, a

1, a

1 , a

1 are computed from the central difference technique and correspond to the values of the

configurational temperatures mentioned in the caption of Fig. S9. The second order derivatives a2, a

2, a

2 , a

2

are zero since the expansion is around the mid-points of the SLRs. The third order coefficients are fitted so

as to achieve a reasonably good fit to the data and are as follows: (1) a3 = 0.012 for mid (2) a

3 = 0.015 for

mid

(3) a3 = 0.018 for

mid(4) a3 = 0.006 for

mid. In all the cases, the fourth and fifth order derivatives

are considered zero.

12

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34 36 38

0.7

0.8

0.9

-1.79

-1.77

-1.75

34 36 38

0.7

0.8

0.9

R SLR

b

[m, m]

[mid]

f4b

f4cb

[m, m]

[mid]

106 MC steps

Fig. S11 The zoomed in portion of the trajectory in Fig. S7. Here b denotes block averages over 91800

MC steps. The other symbols have the same meaning as in Fig. S7

13

0.445

0.449

34 35 36 37 38

0.15

0.19

0.23

0.445

0.449

34 35 36 37 38

0.15

0.19

0.23

R SLR

b

[m, m]

3B4c b

b

3B4c b

106 MC steps

Fig. S12 The zoomed in portion of the trajectory in Fig. S8. Here b denotes block averages over 91800

MC steps. The other symbols have the same meaning as in Fig. S8

14

-1.81

-1.79

-1.77

-1.75

0 2 4 6

0.7

0.75

0.8

0.85

-1.81

-1.79

-1.77

-1.75

0 2 4 6

0.7

0.75

0.8

0.85

-1.81

-1.79

-1.77

-1.75

0 2 4 6

0.7

0.75

0.8

0.85

RSLR

[sm

, sm

]

[smid

]

b

f4b

f4cbb

f4b

f4cb

[sm, sm]

[smid

]

106 MC steps

Fig. S13 The shorter NPT-MC trajectory with N = 4096 particles. The block averages b are over 105 MC

steps. The blue symbols represent configurations corresponding to the local minimum [sm, s

m] of the probability

distribution, which is located within the rectangular area formed by the points (sm /2, s

m /2), and

(sm + /2, s

m + /2), where s

m = 1.754, s

m = 0.44952, = 4.4 104, and = 1.2 104. The

red symbols denotes the configurations along the trajectory with the per particle potential energy () in the

interval [smid] = (s

mid /2, s

mid + /2). The vertical dashed line (dynamical crossover) corresponds to

the point along the trajectory beyond which the potential energy fluctuations are biased towards energies

progressively less than 1.760 0.001.

15

0.447

0.449

0.451

0 2 4 6

0.17

0.21

0.447

0.449

0.451

0 2 4 6

0.17

0.21

RSLR

[sm, sm]

b

3B4c b

b

3B4c b

106 MC steps

Fig. S14 The block averages of the density , and the per particle 3-body energy of largest 4-coordinated

network 3B4c are shown for the same trajectory as in Fig. S13. The block averages are taken over 105 MC steps.

The other symbols have the same meaning as in Fig. S13.

16

0

4

8

12

-1.77 -1.76 -1.75 -1.74

11

12

-1.76 -1.755

log

p(

)+co

nst

.

SLR

final

smid

= 1.7566

R2 = 0.9999999

Tc = 204.37 K

Fig. S15 The potential energy distributions generated by the trajectory in Fig. S13. The x (green) symbols

represents final distribution obtained from trajectory upto 12.2 million MC steps. By final, we mean that the

distribution is not expected to evolve further in the range of values in the above figure.

17

-1.79

-1.77

-1.75

6 7

0.7

0.75

0.8

0.85

-1.79

-1.77

-1.75

6 7

0.7

0.75

0.8

0.85

-1.79

-1.77

-1.75

6 7

0.7

0.75

0.8

0.85

RSLR

[sm

, sm

]

[smid

]

b

f4b

f4cb

106 MC steps

Fig. S16 The zoomed in portion of the trajectory in Fig. S13. The block averages b are over 105 MC

steps. The symbols have the same meaning as in Fig. S13

18

0.447

0.449

6 7

0.17

0.21

0.447

0.449

6 7

0.17

0.21

RSLR

[sm, sm]

b

3B4c b

106 MC steps

Fig. S17 The zoomed in portion of the trajectory in Fig. S14. Here b denotes block averages over 105

MC steps. The other symbols have the same meaning as in Fig. S14