Erasing no-man’s land by thermodynamically stabilizing the liquid-liquid transition intetrahedral particles

Frank SmallenburgDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

Correspondence should be addressed to F. Smallenburg: (f. smallenburg@ gmail. com ).

Laura FilionSoft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science,

Utrecht University, Princetonplein 5, Utrecht, 3584 CC, the Netherlands.

Francesco SciortinoDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

EFFECTS OF THE LIQUID-LIQUID PHASESEPARATION OUTSIDE THE COEXISTENCE

REGION

Although definitive evidence of a liquid-liquid (LL)phase separation requires observation at temperatures atand below the LL critical point (LLCP), its presence af-fects the thermodynamic behaviour of the system even athigher temperatures. In particular, quantities divergingat the critical point exhibit loci of extrema which can beobserved in a wide region of the phase diagram [1] andoriginate at the critical point. Examples are providedby the isothermal compressibility KT and the isobaricheat capacity CP . In the LLCP hypothesis, the densityanomalies of water and other tetrahedral liquids, wherethe density of the liquid phase exhibits a maximum asa function of the temperature at constant pressure, hasbeen related to the extrema in response functions em-anating from the LLCP [2]. In Fig. S1, we draw thepressure-temperature (P − T ) phase diagram of the sys-tem with cosφ = 0.825 and L = 0.5σ, where the LLphase transition is thermodynamically stable. We do ob-serve a line of constant-pressure density maxima ρmax

that occurs at temperatures above the LLCP, as well aslines connecting the extrema in KT and CP . Qualita-tively, the diagram bears a striking resemblence to theone determined for the ST2 model for water[2]. Consis-tent with thermodynamic requirements [3], the point atwhich dP/dT |ρmax

= ∞ is also a KT extremum.The slope of the LL coexistence line Pcoex(T ) is posi-

tive, while it is expected to be negative in ST2. Via theClausius-Clapeyron equation, this slope can be directlyrelated to the difference in entropy between the two co-existing phases:

dPcoex

dT=

∆S

∆V, (1)

where ∆S and ∆V are respectively the difference in en-tropy and volume between the two phases. The posi-tive slope indicates that the entropy of the high-densityliquid (HDL) is lower than that of the low-density liq-

uid (LDL). We also note that by construction, in thismodel, at low T both phases are fully bonded and hencehave the same energy, imposing a vanishing coexistingpressure for T → 0. Possibly, the slope of the coexis-tence curve, expecially close to the critical point, can bemodulated by imposing a repulsive term disfavouring thedenser phase and/or modulating the angular profile of thepotential modeling the bond. Indeed in earlier work, ithas been shown that increasing the bond directionality ofa tetrahedral Kern-Frenkel-based potential, by combin-ing a weaker flexible patch with a stronger narrow patch,can switch the slope of the Widom line (the line of max-ima in CP ) from positive to negative [4]. Interestingly,in our case the slope of the coexistence line already de-creases close to the critical point, and the Widom line,typically seen as the extension of the coexistence line,already shows a negative slope, suggesting that a minormodification to the model may be sufficient to match thisaspect of the ST2 water phase diagram.

POTENTIAL ENERGY IN THE FLUID

Fig. S2 shows the behaviour of the potential energy Uas a function of density for several different bond flexi-bilities (cosφ) and temperatures (T ), at fixed L = 0.5σ.The system exhibits a clear minimum in the potentialenergy at an “optimal” network density [5], a character-istic of network-forming fluids. The minimum originatesfrom the typical monotonic decrease of U with densityat low ρ and from the breaking of the bonds associatedwith the progressive distortion of the tetrahedral geom-etry on increasing ρ beyond the optimal value. Counter-intuitively, the minimum becomes deeper for less flexiblebonds (higher cosφ). Although one might expect thatmaking the bonds more directional would lead to a lowernumber of bonds in the system, the larger entropy of net-work states with broken bonds for wide angles leads toa larger U when cosφ is large. Additionally, less flex-ible bonds also lead to more strongly structured tetra-hedral networks [6], which may favor bond formation.

Erasing no-man’s land by thermodynamically stabilizing the liquid–liquid transition in tetrahedral particles

Erasing no-man’s land by thermodynamically stabilizing the liquid-liquid transition intetrahedral particles

Frank SmallenburgDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

Correspondence should be addressed to F. Smallenburg: (f. smallenburg@ gmail. com ).

Laura FilionSoft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science,

Utrecht University, Princetonplein 5, Utrecht, 3584 CC, the Netherlands.

Francesco SciortinoDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

EFFECTS OF THE LIQUID-LIQUID PHASESEPARATION OUTSIDE THE COEXISTENCE

REGION

Although definitive evidence of a liquid-liquid (LL)phase separation requires observation at temperatures atand below the LL critical point (LLCP), its presence af-fects the thermodynamic behaviour of the system even athigher temperatures. In particular, quantities divergingat the critical point exhibit loci of extrema which can beobserved in a wide region of the phase diagram [1] andoriginate at the critical point. Examples are providedby the isothermal compressibility KT and the isobaricheat capacity CP . In the LLCP hypothesis, the densityanomalies of water and other tetrahedral liquids, wherethe density of the liquid phase exhibits a maximum asa function of the temperature at constant pressure, hasbeen related to the extrema in response functions em-anating from the LLCP [2]. In Fig. S1, we draw thepressure-temperature (P − T ) phase diagram of the sys-tem with cosφ = 0.825 and L = 0.5σ, where the LLphase transition is thermodynamically stable. We do ob-serve a line of constant-pressure density maxima ρmax

that occurs at temperatures above the LLCP, as well aslines connecting the extrema in KT and CP . Qualita-tively, the diagram bears a striking resemblence to theone determined for the ST2 model for water[2]. Consis-tent with thermodynamic requirements [3], the point atwhich dP/dT |ρmax

= ∞ is also a KT extremum.The slope of the LL coexistence line Pcoex(T ) is posi-

tive, while it is expected to be negative in ST2. Via theClausius-Clapeyron equation, this slope can be directlyrelated to the difference in entropy between the two co-existing phases:

dPcoex

dT=

∆S

∆V, (1)

where ∆S and ∆V are respectively the difference in en-tropy and volume between the two phases. The posi-tive slope indicates that the entropy of the high-densityliquid (HDL) is lower than that of the low-density liq-

uid (LDL). We also note that by construction, in thismodel, at low T both phases are fully bonded and hencehave the same energy, imposing a vanishing coexistingpressure for T → 0. Possibly, the slope of the coexis-tence curve, expecially close to the critical point, can bemodulated by imposing a repulsive term disfavouring thedenser phase and/or modulating the angular profile of thepotential modeling the bond. Indeed in earlier work, ithas been shown that increasing the bond directionality ofa tetrahedral Kern-Frenkel-based potential, by combin-ing a weaker flexible patch with a stronger narrow patch,can switch the slope of the Widom line (the line of max-ima in CP ) from positive to negative [4]. Interestingly,in our case the slope of the coexistence line already de-creases close to the critical point, and the Widom line,typically seen as the extension of the coexistence line,already shows a negative slope, suggesting that a minormodification to the model may be sufficient to match thisaspect of the ST2 water phase diagram.

POTENTIAL ENERGY IN THE FLUID

Fig. S2 shows the behaviour of the potential energy Uas a function of density for several different bond flexi-bilities (cosφ) and temperatures (T ), at fixed L = 0.5σ.The system exhibits a clear minimum in the potentialenergy at an “optimal” network density [5], a character-istic of network-forming fluids. The minimum originatesfrom the typical monotonic decrease of U with densityat low ρ and from the breaking of the bonds associatedwith the progressive distortion of the tetrahedral geom-etry on increasing ρ beyond the optimal value. Counter-intuitively, the minimum becomes deeper for less flexiblebonds (higher cosφ). Although one might expect thatmaking the bonds more directional would lead to a lowernumber of bonds in the system, the larger entropy of net-work states with broken bonds for wide angles leads toa larger U when cosφ is large. Additionally, less flex-ible bonds also lead to more strongly structured tetra-hedral networks [6], which may favor bond formation.

Erasing no-man’s land by thermodynamically stabilizing the liquid-liquid transition intetrahedral particles

Frank SmallenburgDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

Correspondence should be addressed to F. Smallenburg: (f. smallenburg@ gmail. com ).

Laura FilionSoft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science,

Utrecht University, Princetonplein 5, Utrecht, 3584 CC, the Netherlands.

Francesco SciortinoDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

EFFECTS OF THE LIQUID-LIQUID PHASESEPARATION OUTSIDE THE COEXISTENCE

REGION

Although definitive evidence of a liquid-liquid (LL)phase separation requires observation at temperatures atand below the LL critical point (LLCP), its presence af-fects the thermodynamic behaviour of the system even athigher temperatures. In particular, quantities divergingat the critical point exhibit loci of extrema which can beobserved in a wide region of the phase diagram [1] andoriginate at the critical point. Examples are providedby the isothermal compressibility KT and the isobaricheat capacity CP . In the LLCP hypothesis, the densityanomalies of water and other tetrahedral liquids, wherethe density of the liquid phase exhibits a maximum asa function of the temperature at constant pressure, hasbeen related to the extrema in response functions em-anating from the LLCP [2]. In Fig. S1, we draw thepressure-temperature (P − T ) phase diagram of the sys-tem with cosφ = 0.825 and L = 0.5σ, where the LLphase transition is thermodynamically stable. We do ob-serve a line of constant-pressure density maxima ρmax

that occurs at temperatures above the LLCP, as well aslines connecting the extrema in KT and CP . Qualita-tively, the diagram bears a striking resemblence to theone determined for the ST2 model for water[2]. Consis-tent with thermodynamic requirements [3], the point atwhich dP/dT |ρmax

= ∞ is also a KT extremum.The slope of the LL coexistence line Pcoex(T ) is posi-

tive, while it is expected to be negative in ST2. Via theClausius-Clapeyron equation, this slope can be directlyrelated to the difference in entropy between the two co-existing phases:

dPcoex

dT=

∆S

∆V, (1)

where ∆S and ∆V are respectively the difference in en-tropy and volume between the two phases. The posi-tive slope indicates that the entropy of the high-densityliquid (HDL) is lower than that of the low-density liq-

uid (LDL). We also note that by construction, in thismodel, at low T both phases are fully bonded and hencehave the same energy, imposing a vanishing coexistingpressure for T → 0. Possibly, the slope of the coexis-tence curve, expecially close to the critical point, can bemodulated by imposing a repulsive term disfavouring thedenser phase and/or modulating the angular profile of thepotential modeling the bond. Indeed in earlier work, ithas been shown that increasing the bond directionality ofa tetrahedral Kern-Frenkel-based potential, by combin-ing a weaker flexible patch with a stronger narrow patch,can switch the slope of the Widom line (the line of max-ima in CP ) from positive to negative [4]. Interestingly,in our case the slope of the coexistence line already de-creases close to the critical point, and the Widom line,typically seen as the extension of the coexistence line,already shows a negative slope, suggesting that a minormodification to the model may be sufficient to match thisaspect of the ST2 water phase diagram.

POTENTIAL ENERGY IN THE FLUID

Fig. S2 shows the behaviour of the potential energy Uas a function of density for several different bond flexi-bilities (cosφ) and temperatures (T ), at fixed L = 0.5σ.The system exhibits a clear minimum in the potentialenergy at an “optimal” network density [5], a character-istic of network-forming fluids. The minimum originatesfrom the typical monotonic decrease of U with densityat low ρ and from the breaking of the bonds associatedwith the progressive distortion of the tetrahedral geom-etry on increasing ρ beyond the optimal value. Counter-intuitively, the minimum becomes deeper for less flexiblebonds (higher cosφ). Although one might expect thatmaking the bonds more directional would lead to a lowernumber of bonds in the system, the larger entropy of net-work states with broken bonds for wide angles leads toa larger U when cosφ is large. Additionally, less flex-ible bonds also lead to more strongly structured tetra-hedral networks [6], which may favor bond formation.

Erasing no-man’s land by thermodynamically stabilizing the liquid-liquid transition intetrahedral particles

Frank SmallenburgDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

Correspondence should be addressed to F. Smallenburg: (f. smallenburg@ gmail. com ).

Laura FilionSoft Condensed Matter and Biophysics, Debye Institute for Nanomaterials Science,

Utrecht University, Princetonplein 5, Utrecht, 3584 CC, the Netherlands.

Francesco SciortinoDepartment of Physics, Sapienza, Universita di Roma, Piazzale Aldo Moro 2, I-00185, Roma, Italy.

EFFECTS OF THE LIQUID-LIQUID PHASESEPARATION OUTSIDE THE COEXISTENCE

REGION

Although definitive evidence of a liquid-liquid (LL)phase separation requires observation at temperatures atand below the LL critical point (LLCP), its presence af-fects the thermodynamic behaviour of the system even athigher temperatures. In particular, quantities divergingat the critical point exhibit loci of extrema which can beobserved in a wide region of the phase diagram [1] andoriginate at the critical point. Examples are providedby the isothermal compressibility KT and the isobaricheat capacity CP . In the LLCP hypothesis, the densityanomalies of water and other tetrahedral liquids, wherethe density of the liquid phase exhibits a maximum asa function of the temperature at constant pressure, hasbeen related to the extrema in response functions em-anating from the LLCP [2]. In Fig. S1, we draw thepressure-temperature (P − T ) phase diagram of the sys-tem with cosφ = 0.825 and L = 0.5σ, where the LLphase transition is thermodynamically stable. We do ob-serve a line of constant-pressure density maxima ρmax

that occurs at temperatures above the LLCP, as well aslines connecting the extrema in KT and CP . Qualita-tively, the diagram bears a striking resemblence to theone determined for the ST2 model for water[2]. Consis-tent with thermodynamic requirements [3], the point atwhich dP/dT |ρmax

= ∞ is also a KT extremum.The slope of the LL coexistence line Pcoex(T ) is posi-

tive, while it is expected to be negative in ST2. Via theClausius-Clapeyron equation, this slope can be directlyrelated to the difference in entropy between the two co-existing phases:

dPcoex

dT=

∆S

∆V, (1)

where ∆S and ∆V are respectively the difference in en-tropy and volume between the two phases. The posi-tive slope indicates that the entropy of the high-densityliquid (HDL) is lower than that of the low-density liq-

uid (LDL). We also note that by construction, in thismodel, at low T both phases are fully bonded and hencehave the same energy, imposing a vanishing coexistingpressure for T → 0. Possibly, the slope of the coexis-tence curve, expecially close to the critical point, can bemodulated by imposing a repulsive term disfavouring thedenser phase and/or modulating the angular profile of thepotential modeling the bond. Indeed in earlier work, ithas been shown that increasing the bond directionality ofa tetrahedral Kern-Frenkel-based potential, by combin-ing a weaker flexible patch with a stronger narrow patch,can switch the slope of the Widom line (the line of max-ima in CP ) from positive to negative [4]. Interestingly,in our case the slope of the coexistence line already de-creases close to the critical point, and the Widom line,typically seen as the extension of the coexistence line,already shows a negative slope, suggesting that a minormodification to the model may be sufficient to match thisaspect of the ST2 water phase diagram.

POTENTIAL ENERGY IN THE FLUID

Fig. S2 shows the behaviour of the potential energy Uas a function of density for several different bond flexi-bilities (cosφ) and temperatures (T ), at fixed L = 0.5σ.The system exhibits a clear minimum in the potentialenergy at an “optimal” network density [5], a character-istic of network-forming fluids. The minimum originatesfrom the typical monotonic decrease of U with densityat low ρ and from the breaking of the bonds associatedwith the progressive distortion of the tetrahedral geom-etry on increasing ρ beyond the optimal value. Counter-intuitively, the minimum becomes deeper for less flexiblebonds (higher cosφ). Although one might expect thatmaking the bonds more directional would lead to a lowernumber of bonds in the system, the larger entropy of net-work states with broken bonds for wide angles leads toa larger U when cosφ is large. Additionally, less flex-ible bonds also lead to more strongly structured tetra-hedral networks [6], which may favor bond formation.

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3030

NATURE PHYSICS | www.nature.com/naturephysics 1

© 2014 Macmillan Publishers Limited. All rights reserved.

2

Ρmax

CPmax

KTmax

CPmin

KTmin

0.00 0.02 0.04 0.06 0.08 0.10 0.120.000

0.002

0.004

0.006

0.008

kBT � Ε

PΣ

3�

Ε

FIG. S1: Phase diagram of the tetramer model with cosφ =0.825 and L = 0.5σ in the P−T plane. The blue line shows theLL coexistence line, ending in a critical point. The black linedenotes the line of density maxima, where

(∂ρ∂T

)P

= 0. Thegreen line connects extrema in the isothermal compressibility

KT , i.e. points where(∂KT∂T

)P

= 0, with the thicker solid line

indicating maxima and the dashed line indicating minima.Similarly, the red line traces the extrema in the isobaric heat

capacity CP , i.e. where(∂CP∂P

)T

= 0.

A secondary minimum could be expected at a densityapproximately a factor of two larger than the optimaldensity, indicating the possible formation of an interpen-etrated network. Between these two densities, U showsa region of negative curvature, indicating an energeticdriving force favoring phase separation [7].

Figure S3 shows U as a function of the reduced den-sity ρ∗ = ρ(σ+L+ δ/2)3 for three different values of thesoftness. As expected, increasing the length of the armsleads to an overall lower number of bonds in the system.Additionally, the increase in energy when compressingthe liquid beyond its optimal density is much less pro-nounced for small L, which is consistent with the factthat the LL phase transition disappears for sufficientlysmall L.

Finally, Fig. S4 shows the effects of cosφ and L on thepotential energy at fixed T .

COMMON TANGENTS

Figure S5 shows the common tangent constructions attemperature kBT/ε = 0 (with kB Boltzmann’s constant)for three different values of the bond flexibility at fixedsoftness L = 0.5σ.

HISTOGRAMS

To determine phase coexistences between the variousfluid phases, we use both successive umbrella sampling(SUS) simulations (in the grand canonical ensamble) and

aL cos Φ = 0.8

0.07 0.08 0.09 0.10 0.11 0.12- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

ΡΣ3

U�N

Ε

kB T� Ε = 0.06

kB T� Ε = 0.07

kB T� Ε = 0.08

kB T� Ε = 0.09

bL cos Φ = 0.825

0.07 0.08 0.09 0.10 0.11 0.12- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

ΡΣ3

U�N

Ε

cL cos Φ = 0.9

0.07 0.08 0.09 0.10 0.11 0.12- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

ΡΣ3

U�N

Ε

FIG. S2: Potential energy U per particle as a function ofthe density ρ at fixed L = 0.5σ, for three different valuesof the bond flexibility cosφ and a range of temperatures, asindicated. In each plot, the set of temperatures shown is thesame. The lines are fits to data from Monte Carlo simulations.In panel (c), the raw simulation data is also shown as points.

free-energy calculations, as explained in the Methods sec-tion. When using the SUS simulations, we evaluate thehistogram P (N), the probability to find N particles inthe simulation box at fixed T , volume V and chemicalpotential µ. The histogram can be reweighted analyti-cally by changing µ. In case of a coexistence, P (N) mustshow two peaks of equal area, centered around the coex-istence densities. If a coexistence is absent, only a single

© 2014 Macmillan Publishers Limited. All rights reserved.

3

aL L = 0

0.7 0.8 0.9 1.0 1.1 1.2 1.3- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

-1.70

Ρ*

U�N

Ε

kB T� Ε = 0.07

kB T� Ε = 0.08

kB T� Ε = 0.09

kB T� Ε = 0.1

bL L = 0.3Σ

0.7 0.8 0.9 1.0 1.1 1.2 1.3- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

-1.70

Ρ*

U�N

Ε

cL L = 0.5Σ

0.7 0.8 0.9 1.0 1.1 1.2 1.3- 2.00

-1.95

-1.90

-1.85

-1.80

-1.75

-1.70

Ρ*

UN

Ε

FIG. S3: Potential energy U per particle as a function of thedensity ρ at fixed cosφ = 0.9, for three different values of thesoftness L and a range of temperatures, as indicated. In eachplot, the set of temperatures shown is the same. The lines arefits to data from Monte Carlo simulations. Note that panel(c) is partly based on the same data as Fig. S2c.

peak is observed. At low temperatures, the SUS simu-lations become difficult to equilibrate, and we resort tothermodynamic integration to determine the coexistencedensities instead. To obtain the Helmholtz free energyper particle (f = F/N) as a function of T , we use:

β2f(ρ, T2) = β1f(ρ, Tref) +

∫ β2

βref

dβ u(ρ, T ), (2)

0.06 0.08 0.10 0.12- 2.00

-1.95

-1.90

-1.85

Ρ*

U�N

Ε cos Φ = 0.9cos Φ = 0.875cos Φ = 0.85cos Φ = 0.825cos Φ = 0.8cos Φ = 0.075

0.7 0.8 0.9 1.0 1.1- 2.00

-1.95

-1.90

-1.85

Ρ*

U�N

Ε

L = 0L = 0.05ΣL = 0.1ΣL = 0.2ΣL = 0.3ΣL = 0.4ΣL = 0.5Σ

FIG. S4: Top: potential energy as a function of density atfixed arm length L = 0.5σ and temperature kBT/ε = 0.08,for different values of the bond flexibility cosφ. Bottom: po-tential energy as a function of density at fixed bond flexibilitycosφ = 0.9 and temperature kBT/ε = 0.08, for different val-ues of the arm length L. The lines are fits to the simulationdata.

where u(ρ, T ) the average potential energy per particleof the system at a given density and temperature, andβ = 1/kBT .

To calculate a reference Helmholtz free energy at hightemperature Tref , we perform another run of SUS simu-lations, evaluating the chemical potentials in each of theSUS windows µ

N,N+1as

βµN,N+1

= βµ− log

(P (N + 1)

P (N)

),

with µ the chemical potential used in the grand-canonicalSUS simulation, and P (N) the probability of observingN particles in the simulation. We also perform an NPTsimulation at the same Tref and intermediate density. Wethen combine the pressure P of the NPT simulation withµ

N,N+1obtained from the SUS simulation at the same

density (ρref) to obtain a free energy

fref = µ− P/ρref . (3)

Next, we use µN,N+1

from all SUS windows to evaluatethe free energy at Tref for different numbers of particles

© 2014 Macmillan Publishers Limited. All rights reserved.

4

Diamond

Gas

LDL HDL

BCC

aL cos Φ = 0.825

0.00 0.05 0.10 0.15-0.08

-0.06

-0.04

-0.02

0.00

ΡΣ3

HΒf

+cL

ΡΣ

3

Gas

LDL

Diamond

HDL

BCC

bL cos Φ = 0.8375

0.06 0.08 0.10 0.12 0.14 0.16

- 0.07

- 0.06

- 0.05

- 0.04

- 0.03

ΡΣ3

HΒ

f+

cL

ΡΣ

3

LDL

Diamond

Gas

BCC

cL cos Φ = 0.9

0.00 0.05 0.10 0.15

- 0.010

- 0.005

0.000

0.005

ΡΣ3

HΒ

f+

cL

ΡΣ

3

FIG. S5: Common tangent plots at kBT/ε = 0 for three ofthe phase diagrams in the Fig. 4 of the main text. Thethick lines indicate the free energy per unit volume of thedifferent phases. At each density, the phase or linear combi-nation of two phases with the lowest free energy is stable. Aconstant linear term (cρσ3) has been added in each plot toenhance readability; this has no effect on the resulting phasebehaviour. Metastable phases are shown as dotted lines. Thethinner red lines show the common tangent constructions con-necting coexisting state points.

N in the same volume:

F (N,V, T ) = Nfref +

∫ N

Nref

µN,N+1

dN, (4)

where Nref = V ρref is the number of particles in thesystem at the reference density. We then evaluate, for

each desired density, the free energy at other T using Eq.2. Finally, to calculate a histogram from this free energy,we use:

P (N) ∝ exp(βµN) exp(−βF (N,V, T )), (5)

where µ is again tuned such that the two peaks haveequal area, and the volume is chosen to match the oneused in the SUS simulations.

Figure S6 shows examples of histograms at cosφ = 0.9as obtained from SUS simulations, and histograms calcu-lated (at the same volume V = 1728σ3) from the free en-ergies obtained via thermodynamic integration over thetemperature. While the SUS data shows more noise, thecoexistence densities from both methods agree very well.

Figure S6-(c) shows SUS histograms for both the gas-liquid and the LL transition for cosφ = 0.9 and kBT/ε =0.08 and kBT/ε = 0.09. Note that the liquid coex-isting with the gas and the low density liquid coexist-ing with the high density liquid have different densities.The chemical potentials of the two transitions are dis-tinct as well: for kBT/ε = 0.08, βµGL = −12.87 andβµLL = −11.16, and for kBT/ε = 0.09, βµGL = −10.03and βµLL = −8.64.

A second set of histograms, again at V = 1728σ3, areshown in Fig. S7, for cosφ = 0.825. Interestingly, thevalley between the two phases is not very deep even atT = 0. In other words, it is still easy for the system toswitch between the two phases. Note that critical pointsare estimated by examining at which T the two peaksin the P (N) histogram were twice as high as the valleybetween them [8].

EFFECT OF SYSTEM SIZE

To ensure that the phase transition is not an artifactof our small system size, we performed additional SUSsimulations for larger system sizes, with a volume up toeight times larger than the standard one. In particu-lar, we have investigated volumes up to V = 13824σ3 forL = 0.5σ and cosφ = 0.9. The LL critical point is clearlypresent for all explored system sizes, confirming the inde-pendence on system size of the critical phenomena. Thehistograms for several T for the largest system size inves-tigated, calculated implementing histogram reweighting,are shown in Fig. S8.

COMPARING THE MODEL TO WATER

Qualitatively, the structure of liquid water at low tem-peratures is similar to that of tetravalent patchy particleswith strong bond rigidity [6]. Thus, despite the simplic-ity of our model, it is instructive to consider mapping thetetramer model introduced here onto water. To do this,

© 2014 Macmillan Publishers Limited. All rights reserved.

5

120 140 160 180 200 220 2400.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

N

PHN

L kB T� Ε = 0.1kB T� Ε = 0.095kB T� Ε = 0.09kB T� Ε = 0.08kB T� Ε = 0.07aL

120 140 160 180 200 220 2400.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

N

PHN

L kB T� Ε = 0.1kB T� Ε = 0.095kB T� Ε = 0.09kB T� Ε = 0.08kB T� Ε = 0.07bL

0 50 100 150 200 2500.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

N

PHN

L

120 140

cL

kB T� Ε = 0.09: LL

kB T� Ε = 0.09: GL

kB T� Ε = 0.08: LL

kB T� Ε = 0.08: GL

FIG. S6: Histograms obtained from a) SUS simulations andb) free-energy calculations (middle), at L = 0.5σ, cosφ = 0.9,and a range of temperatures as indicated. Note that for read-ibility, only some of the points are denoted by symbols. Inc) we replot two of the SUS histograms, and include the SUShistograms for the gas-liquid coexistences at the same temper-ature (but different chemical potential) to provide evidencethat the liquid coexisting with the gas and the low densityliquid coexisting with the high density liquid have differentdensities (as clarified in the inset, which shows a zoomed-inversion of the region outlined by dashed lines). Note that thegas peaks are barely visible, being located at N ≈ 0.

140 160 180 200 220 2400.00

0.01

0.02

0.03

0.04

N

PHN

L kB T� Ε = 0.09kB T� Ε = 0.08kB T� Ε = 0.07kB T� Ε = 0.065kB T� Ε = 0.06

120 140 160 180 200 220 2400.00

0.01

0.02

0.03

0.04

N

PHN

L

kB T� Ε = 0.09kB T� Ε = 0.08kB T� Ε = 0.07kB T� Ε = 0.06kB T� Ε = 0.

FIG. S7: Histograms obtained from SUS simulations (top)and free-energy calculations (bottom), at L = 0.5σ, cosφ =0.825, and a range of temperatures as indicated. Note that forreadibility, only some of the points are denoted by symbols.

1100 1200 1300 1400 1500 1600 17000.000

0.005

0.010

0.015

N

PHN

L

kB T� Ε = 0.918kB T� Ε = 0.91kB T� Ε = 0.901kB T� Ε = 0.892

FIG. S8: Histograms obtained from SUS simulations for alarge system size (V = 13824σ3), at L = 0.5σ, cosφ = 0.9,and a range of temperatures as indicated.

we require an estimation of the bond flexibility cosφ andbond length L.

From comparisons between the angular distributionand strength of the pre-peak of the structure factor be-tween water and a standard Kern-Frenkel model, wecan conclude that typical water models have a bonds

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6

for which the directionality roughly corresponds to apatch opening angle of cos θ ≈ 0.95 in the Kern-Frenkelmodel [6]. This is exactly the value of the opening an-gle cos θ for bonding used in this article when no extraflexibility is added via cosφ. Thus, in terms of flexibility,the results for cosφ = 1 (region III in Fig. 3 of the mainarticle) refer to the case of water.

Concerning L, we recall that this quantity controls theratio between the distance of closest approach for un-bonded (determined by σ) and bonded pairs (determinedby σ+L) of particles. Analysis of the radial distributionfunction (and of its density dependence) can thus provideestimates of L for atomic and molecular systems. For thecase of water, even when the density varies from ρ = 0.92to ρ = 1.4 g/cm3, there is no indication of pair of parti-cles located at distances closer than the typical hydrogenbond distance: no signal shows up in the radial distri-bution function before the main peak [9]. Hence, onecan conclude that water should be mapped into the caseL → 0 and cosφ → 1. Consistent with experimental re-sults, our calculations confirm that both for L → 0 andcosφ → 1 spontaneous crystallization prevails (Figs. 2and 3 in the main text).

[1] Xu, L. et al. Relation between the widom line and the dy-namic crossover in systems with a liquid–liquid phase tran-

sition. Proc. Natl. Acad. Sci. U.S.A. 102, 16558–16562(2005).

[2] Poole, P. H., Saika-Voivod, I. & Sciortino, F. Densityminimum and liquid–liquid phase transition. J. Phys.:Condens. Matter 17, L431 (2005).

[3] Sastry, S., Debenedetti, P. G., Sciortino, F. & Stanley,H. E. Singularity-free interpretation of the thermodynam-ics of supercooled water. Phys. Rev. E 53, 6144 (1996).

[4] Tu, Y., Buldyrev, S. V., Liu, Z., Fang, H. & Stanley,H. E. Different water scenarios for a primitive model withtwo types of hydrogen bonds. Europhys. Lett. 97, 56005(2012).

[5] De Michele, C., Tartaglia, P. & Sciortino, F. Slow dynam-ics in a primitive tetrahedral network model. J. Chem.Phys. 125, 204710 (2006).

[6] Saika-Voivod, I., Smallenburg, F. & Sciortino, F. Under-standing tetrahedral liquids through patchy colloids. J.Chem. Phys. 139, 234901 (2013).

[7] Poole, P. H., Sciortino, F., Essmann, U. & Stanley, H. E.Spinodal of liquid water. Phys. Rev. E 48, 3799 (1993).

[8] Wilding, N. B. Critical-point and coexistence-curve prop-erties of the lennard-jones fluid: a finite-size scaling study.Phys. Rev. E 52, 602 (1995).

[9] Starr, F. W. et al. Thermodynamic and structural aspectsof the potential energy surface of simulated water. PhysicalReview E 63, 041201 (2001).

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