Table Of ContentLiquid-liquid phase transition model incorporating evidence for ferroelectric state
near the lambda-point anomaly in supercooled water.
P.O. Fedichev
Quantum Pharmaceuticals Ltd, Ul. Kosmonavta Volkova 6a-1205,
125171, Moscow, Russian Federation, http://q-pharm.com∗
L.I. Menshikov
NRC Kurchatov Institute, Kurchatov Square 1, 123182, Moscow, Russian Federation
Weproposeaunifiedmodelcombiningthefirst-orderliquid-liquidandthesecond-orderferroelec-
tric phase transitions models and explaining various features of the λ-point of liquid water within
a single theoretical framework. It becomes clear within the proposed model that not only does the
2 long-rangedipole-dipoleinteractionofwatermoleculesyieldalargevalueofdielectricconstant(cid:15)at
1 room temperatures, our analysis shows that the large dipole moment of the water molecules also
0 leads to a ferroelectric phase transition at a temperature close to the λ−point. Our more refined
2 model suggests that the phase transition occurs only in the low density component of the liquid
n and is the origin of the singularity of the dielectric constant recently observed in experiments with
a supercooled liquid water at temperature T ≈233K. This combined model agrees well with nearly
J every available set of experiments and explains most of the well-known and even recently obtained
0 results of MD simulations.
3
Supercooled water exhibits a number of well-known pacity and thermal conductivity [38] peak around T
] λ
t anomalies near the so-called λ−point at temperature as well and that the liquid shows a good deal of or-
f
o T ≈ 228K (at normal pressure), where a good number dering in the vicinity of the λ−point. Similarly, re-
λ
s ofthermodynamicquantitiessuchasheatcapacity,com- cent measurements of the dielectric properties of liquid
.
t pressibility, thermal expansivity, and dielectric constant water confined in nanopores and hence prevented from
a
m all exhibit nearly singular behavior [1–3]. The intrinsic freezing well below the natural freezing point [39] mani-
thermodynamic instability of liquid water at tempera- fest a profound bump in the dielectric constant near the
-
d tures well below the freezing point has been a major ob- λ−point [40, 41]. These observations paint a richer pic-
n stacle both in experimental studies and theoretical mod- ture than a mere first-order liquid-liquid phase transi-
o eling(seee.g. [4–11]forareview). Theobservedfeatures tion and, in fact, bring back an old idea [3] relating the
c are often weak, which suggests a thermodynamic conti- weak singularity of the dielectric constant to a ferroelec-
[
nuity of the various water states near the λ−point and tric phase transition (FPT). Remarkably, the hypothesis
1 relates the observed “singularity” with essentially a ran- was put forward immediately after the discovery of the
v dom sharp feature, a function of the parameters of the λ−point, though the weakness of the observed singular-
3 liquid [12, 13]. A considerably more accepted view is to ities prompted the authors [3] to reject the explanation.
9
ascribethefeaturescharacteristicofaphase-transitionto Furthermore, the ferroelectric instability for a model liq-
1
a first-order liquid-liquid phase transition (LLPT). This uidwithparameterssimilartowaterispredictedtooccur
6
model predicts the existence of a second critical point at a very high and essentially unreachable temperature,
.
1
of water at the temperature T ∼ 200K and pressure ∼ 1200K [42]. A more sophisticated model [43–45] pre-
0 CR
∼ 1kbar [14, 15]. The sharp temperature dependencies dicts a ferroelectric phase transition at a temperature
2
observed near T are attributed to crossing the Widom independentofthedetailsoftheshort-rangeinteractions
1 λ
: line [16], where the density and entropy fluctuations are between the molecules,
v
large [17] and which happens at T ≈T at normal pres-
i λ 4πn d2
X sure. The view is supported by numerous molecular dy- T = 0 0 =210÷236K ≈T , (1)
F 9(cid:15) λ
r namics(MD)simulationsbasedonrealisticwatermodels ∞
a [14, 18–26], simplified analytical models [27–31], and ex- where n is the density and d are the static dipole mo-
0 0
perimental studies [4, 32–34]. The reported anomalies ments of the molecule comprising the liquid. The dielec-
are not restricted to static features, the dynamic proper- tricconstant(cid:15) isnotassociatedwiththemolecules’de-
∞
ties such as the Einstein relation between diffusion and grees of orientational freedom and comes from electron
mobility coefficients [35, 36] and the Arrhenius behavior shell polarization, (cid:15) =4÷5.5 (as discussed in e.g. [46–
∞
oftheliquid’sdynamicproperties[16,36,37]breakdown 48]). Despite bringing the phase transition temperature
near Tλ as well. into the right range, the model predicts a behavior of
Recent MD studies have demonstrated that heat ca- the dielectric constant with changes in temperature that
is far too gradual when compared with empirical results
[40, 41]. Consequently, it cannot even qualitatively ex-
plain all of the features of supercooled water by itself.
∗Electronicaddress: [email protected] Realistic liquids such as water are far more complicated
2
thanamodelpolarliquidconsistingofpointdipoles. For
example, the tetrahedral geometry of H O molecule and
2
its electron shell leads to a polyamorphism phenomenon
[4, 7, 49], namely two or more phases of the same liquid
existing in a mixture at the same time.
To unify the observed ferroelectric-like properties of
water-moleculeorderingandthesingularityofthedielec-
tricconstantnearT [40,41]withthepreviouslyreported
λ
signaturesoftheLLPTwithinasingletheoreticalframe-
work,wecombineoursimplepolarliquidphenomenology
[40, 43–45] with the LLPT hypothesis [4, 14, 15] using a
two-component mixture model of water [50]. We assume
that the equilibrium state of supercooled water is a mix-
ture of macroscopically-sized clusters of the two types:
low density (LDL) and high density liquid (HDL).The
LDLlocallatticeissofterthanthatofHDLandtheden-
Figure 1: Model PT diagram of water (see the explanations
sity of HDL exceeds that of LDL by ∼ 20% [4, 51–55].
in the text).
Since LDL is “softer”, the molecules of the LDL rotate
more or less freely, whereas in the HDL the rotations
are more difficult. This explains why HDL has no ferro-
Here G is the free energy of the HDL component,
electric state at any temperature. We assume that the HDL
and the parameter U characterizes the “energy of mix-
ferroelectric ordering and the FPT apparently observed
ing”. The equilibrium free energy and the LDL frac-
in supercooled liquid bulk water occurs in the LDL com-
tion c are found through minimization of G(c) over c,
ponent only.
G(cid:48)(c)≡[∂G(c,P,T)/∂c] =0, or
The Gibbs free energy of an LDL cluster at a given P,T
pressureP isthesumofthecontributionsfromthepolar (cid:18) (cid:19)
c
lGiqLuid,.GAPLtDhLig,hantdemthpeerlaattutricees,, GTLL>DLT: G,LthDeLe=quGiliPLbDriLum+ (cid:52)G(P,T)+U(1−2c)+RT log 1−c =0, (5)
LDL F
state of LDL corresponds to the disordered paraelectric
where (cid:52)G(P,T)=G −G . The temperature of
phase, whereas at lower temperatures, T < T , LDL LDL HDL
F the second critical point in the model is T = U/(2R)
undergoesasecond-orderphasetransitionandentersthe CR
[50].
long-range-ordered ferroelectric state. Near the phase
To analyze these equations, we follow [50, 57] and as-
transition, where τ =(T −T )/T (cid:28)1, the free energy
F F sumethatthethermodynamicquantitiesin(cid:52)G(P,T)=
of LDL takes the form:
(cid:52)E0 − T(cid:52)S0 + P(cid:52)V0 are practically temperature-
G ≈−Dτ2θ(−τ)+GL (T,P), (2) independent:
LDL LDL
where D ∼ V0n20d20 ∼ 150cal/mol, and V0 = NA/n0 ≈ (cid:52)E0(P,T),(cid:52)S0(P,T),(cid:52)V0(P,T)≈const. (6)
22cm3 is the molar volume of LDL. Since the LDL den-
sity differs by no more than 20% from the total liquid The PT diagram suggested by this model is shown
density, wewillnotdistinguishbetweentheLDLdensity in Fig.1. The AKB line in the Figure is the spin-
and n0. The second-order FPT in LDL manifests itself odal line corresponding to G(cid:48)(c) = 0, G(cid:48)(cid:48)(c) ≡
asasingularityofthedielectricconstant(cid:15)(seee.g. [56]): (cid:2)∂2G(c,P,T)/∂c2(cid:3) = 0. The KB line is the LDL
P,T
spinodal, where the LDL phase loses its thermodynamic
(cid:15)=(cid:15) (1+f(T)), (3)
∞
stability (the local minimum of the function G(c) corre-
where f(T) = 3T /(T −T ) at T > T and f(T) = sponding to the LDL phase disappears on this line). In
F F F
3T /2(T −T) at T < T , which is a much weaker de- turn, the KA line is the HDL spinodal. The Gibbs po-
F F F
pendence than that observed experimentally [40]. The tentialG(c)hasasingleminimumeverywhereabovethe
discrepancy is apparently due to the very sharp temper- AKB line and two minima below this line, at c=c and
1
aturedependenceoftheLDLfractioncnearT ≈T . To c = c . The section KC of the straight line WC is the
λ 2
seethat, wefollow[50]andformulateatwo-liquidmodel liquid-liquid first-order phase transition line correspond-
representing the Gibbs energy of water as the energy of ing to the phase equilibrium conditions: G(c )=G(c ),
1 2
a two-liquid mixture of macroscopic clusters using a rep- G(cid:48)(c ) = G(cid:48)(c ) = 0. According to Eqs.(4) and (5),
1 2
resentation similar to that used in the physics of binary these conditions are equivalent to (cid:52)G(P,T)=0, which,
alloys (see e.g. [57]): according to approximation (6), means that T is a linear
function of P on the KC line, which means that the KC
G(c)=cG +(1−c)G +Uc(1−c)+
LDL HDL lineisitselfastraightlineinFigure1. Thelowandhigh
density amorphous ice regions are denoted by LDA and
+RT [clogc+(1−c)log(1−c)]. (4) HDA.
3
Figure 3: Theoretical temperature dependence of the LDL
fraction (solid lines, see the explanations in the text) versus
empirical data [33] (dots).
Figure 2: Calculated water heat capacity (formulas (7), (8),
(10); solid curve) versus MD calculations [38] (points) and
Eq. (9)toanalyzetheheatcapacitycalculatede.g. using
empirical data [1] (crosses).
MD from [38] and extract the model parameters:
(cid:52)E0 =−860cal·mol−1, (cid:52)S0 =−3.5cal·mol−1K−1,
The KD line in Figure 1 corresponds to the limit
(10)
(cid:52)S → 0 at T → 0 as required by the third law of ther-
and U = 880cal ·mol−1. These parameters correspond
modynamics and which is never the case under the as-
to T = 245K, T = 220K, and P = 1kbar.
sumptions (6). WK is the Widom line, defined by the W CR CR
conditions G(cid:48)(c) = 0 and G(cid:48)(cid:48)(cid:48)(c) = 0, or, equivalently, To calculate PCR we used (cid:52)V0 ≈ 3.8cm3mol−1 from
[50], where very similar parameters were obtained: U ≈
c = 1/2. At small pressures the Widom temperature is
T =(cid:52)E0/(cid:52)S0. Theheatcapacity, C ,consistsofthe 900cal·mol−1, (cid:52)E0 = −250cal·mol−1, (cid:52)S0 = −1cal·
W P
mol−1K−1, and T = 225K. These “original” pa-
two parts CR
rameters yield smaller critical pressure value: P ≈
CR
0.33kbar. TheheatcapacityC calculatedwiththehelp
P
(cid:34)(cid:18) (cid:19) (cid:35)2 of Eqs. (7), (8) and (10) is plotted on Fig.2 against the
∂S
C =T /G(cid:48)(cid:48)(c)=CP +CL, (7) experimental values from [1] and the recent MD calcula-
P ∂c P P
P,T tions [38]. Note that although the MD calculation does
not provide a full match with the experimental curve
where CL is the lattice contribution, which can be in absolute terms, both data sets consistently describe
P
calculated e.g. using Debye approximation, CL = thesamefeatureandhenceapparentlythesamephysics.
P
(cid:104) (cid:105)
C∞(T/Θ)3/ 1+(T/Θ)3 , where C∞ = 18cal · This in combination with the water molecules ordering
P P
predicted by the model (the entropy of LDL is less than
mol−1K−1 (as suggested in [1]) and Θ ≈ 150K is the
that of HDL, (cid:52)S0 <0, in accordance with [6]) and con-
Debye frequency of water. Within the described model
firmed by the MD calculations is the indication of ferro-
the polar part of CP is
P electric transition implicitly present already in the simu-
(cid:2)(cid:52)E0+P(cid:52)V0+U(1−2c)(cid:3)(cid:104)(cid:52)S0−Rlog(cid:16) c (cid:17)(cid:105) lation [38].
CP ≈ 1−c . Once the model parameters (10) are established we
P RT −2U can verify the consistency of the model by observing the
c(1−c) temperature dependence of the LDL fraction c(T) given
(8)
by Eq. (5) (see solid line 1 in Figure 3) and compare it
At small pressures along the Widom line c≈1/2, G(cid:48)(cid:48)(c)
with the empirical data [33] (the dots in the same Fig-
issmall,thefluctuationsarestrong,andthetemperature
ure). There is a qualitative agreement at least at suffi-
dependence of the heat capacity contribution
ciently large temperatures above 200K. We note, that
R(cid:52)2 (cid:52)G(P,T =0) < 0 as T → 0 and therefore, at suffi-
CPP ≈ (cid:104)(T −TW)2+δ2(cid:105), (9) cinietnotlLyDlLo,wi.tee.mcpe→rat1uraess Tall→of0t.heThliiqsuliedadsshouusldtotubren-
lieve that the equilibrium composition of water was not
takesastandardLorenzform,where(cid:52)=T −T ,δ = actuallyachievedintheexperimentsof[33],evenatvery
W CR
(cid:112)
2R(cid:52) (cid:52)/((cid:52)E0(cid:52)S0). The quantity peaks at T = T low temperatures. This may well be due to the increase
W
inagreementwithexperiments[5,6],earlierexplanations of equilibration time at extremely low temperatures, as
[16], and recent calculations [38]. Therefore we can use discussed in e.g. [58, 59]. With this in mind, we can at-
4
means that including more liquid states in the analy-
sis clearly improves the agreement with experimental re-
sults. Below the transition point to the left of the peak,
at T <T , experiments yield distinctly different results.
F
There may be many reasons for that. For example it is
quite possible there is more than one LDL state of water
belowT . ThetemperatureT itselfmaydependonthe
F F
LDL cluster size, and therefore we may face a “continu-
ousset” offerroelectrictransitionsinamultitudeofLDL
forms as the temperature decreases.
Whencomparedwithavailableempiricaldataandnu-
merical calculations, the model calculations of the heat
capacity, LDL fraction, and the dielectric constant sup-
port a broader view, implying that supercooled water is
Figure 4: Dielectric constant for bulk water versus tempera-
indeed a mixture of at least two different components,
ture. Results of formulas (12), (10) are given by solid curve.
namely LDL and HDL. There is a growing body of ev-
Experimentalresults[40,41]arepresentedbypoints,crosses
idence for ferroelectric transition in LDL clusters. In-
correspond to the measurements [3].
dependent of the theoretical arguments given above, the
FPT in LDL can be supported by the structural similar-
tempt to match the LDL fraction measurements only at itybetweenthelocalstructureofLDLandthecrystalline
higher temperatures and obtain better agreement with lattice of stable ice Ih [61, 62]. At normal pressure the
theexperimentusingslightlydifferentmodelparameters paraelectriciceIhenterstheferroelectricstate,iceXI,at
temperature T = 72K [63–66]. The ferroelectric state,
(cid:52)E0 =−920cal·mol−1, (cid:52)S0 =−4cal·mol−1K−1, iceXI,isobtainedfromhexagonaliceIhafterreconstruc-
(11) tion of the crystalline lattice, namely, by the shifting of
and U = 900cal · mol−1 from [50] (see solid line 2 in water molecules in each elementary cell of the crystal. It
Figure 3). Although curves 1 and 2 are clearly different, is interesting to note that theory [67] predicts antiferro-
bothsets,(10)and(11),areveryclosenumerically,which electric ordering of molecular dipole moments for Ih ice
indicates further difficulties in refining the parameters with undeformed crystalline lattice, but the ferroelectric
duetoaverysharpdependenceoftheLDLfractioncnext statewithadeformedlatticehasalowerfreeenergy. Itis
to the Widom line and clear experimental difficulties. not surprising that, due to a less dense local crystalline
An analysis of the heat capacity and the LDL frac- lattice of LDL, the transition temperature observed in
tion measurement let us verify the model and deter- [40, 41] essentially exceeds the corresponding value of
mine the model parameters. Consequently, we may use FPT “ice Ih(cid:10)ice XI”. It is also worth mentioning here
the model to predict the temperature dependence of that there have been important studies concerning the
the dielectric constant and compare it with empirical ferroelectric states of metastable ice Ic, with a cubic lat-
data. Accordingly, the dielectric constant of the liq- tice first predicted in [68] and detected experimentally
uid is the sum of the LDL, (cid:15) = (cid:15) (1+f(T)) with in [69, 70]. The reported explanation of the “ice Ih(cid:10)ice
LDL ∞
f(T) from Eq. (3), and the HDL contributions. The XI” transition gives us cause to believe that the ferro-
HDL fraction is 1 − c, and its dielectric constant can electric phase transition in LDL is also accompanied by
be described by the Debye-Onsager model [60]: (cid:15) = the reconstruction of the local crystalline lattice, i.e. the
HDL
(cid:15) +2πn d2((cid:15) +2)2/(9T). Therefore the full expres- dipolemoment orientation of molecules is strongly linked
∞ 0 0 ∞
sion (cid:15)=c(cid:15) +(1−c)(cid:15) becomes to the lattice’s degrees of freedom. On the other hand,
LDL HDL
FPT occurs only in LDL. Therefore, it is reasonable to
2πn d2 assumethattheKClineinFigure1isinfactthebound-
(cid:15)=(cid:15) [1+c(T)f(τ)]+[1−c(T)] 0 0 ((cid:15) +2)2.
∞ 9T ∞ ary between the ferroelectric and paraelectric regions of
(12) the supercooled water. Here |(cid:52)G |∼|(cid:52)G | and there-
P L
The predicted temperature dependence calculated using fore, the molecular orientation and the lattice’s degrees
the parameters (10) and the value (cid:15) =4.7 correspond- offreedomareequallyimportant“drivingforces” ofboth
∞
ing to T =233K is compared with recent experimental the ferroelectric and the liquid-liquid phase transitions.
F
measurements [3, 40, 41] in Figure 4. At high tempera- Above the FK line the situations is different; there typi-
tures, T >T ,T the model dependence is not far from cally|τ|(cid:28)1andEq. (2)yields|(cid:52)G |(cid:28)|(cid:52)G |. There-
F W P L
the experimentally observed values. The temperature fore, the lattice degrees of freedom dominate. Hence, we
dependence is now sharp, much stronger than that pre- believe that supercooled water is ferroelectric below the
dicted by a simpler, single-component model [40]. This FKP line.
5
[1] C. Angell, J. Shuppert, and J. Tucker, The Journal of [30] M. Girardi, A. Balladares, V. Henriques, and M. Bar-
Physical Chemistry 77, 3092 (1973) bosa,TheJournalofchemicalphysics126,064503(2007)
[2] R. Speedy and C. Angell, The Journal of Chemical [31] K.Stokely,M.Mazza,H.Stanley,andG.Franzese,Pro-
Physics 65, 851 (1976) ceedings of the National Academy of Sciences 107, 1301
[3] I.HodgeandC.Angell,TheJournalofChemicalPhysics (2010)
68, 1363 (1978) [32] O. Mishima, Physical review letters 85, 334 (2000)
[4] O. Mishima and H. Stanley, Nature 396, 329 (1998), [33] F. Mallamace, C. Corsaro, M. Broccio, C. Branca,
ISSN 0028-0836 N. González-Segredo, J. Spooren, S. Chen, and H. Stan-
[5] P. Debenedetti, Journal of Physics: Condensed Matter ley, Proceedings of the National Academy of Sciences
15, R1669 (2003) 105, 12725 (2008)
[6] P. Debenedetti and H. Stanley, Physics Today 56, 40 [34] O. Mishima et al., The Journal of chemical physics 133,
(2003) 144503 (2010)
[7] C. Angell, Annu. Rev. Phys. Chem. 55, 559 (2004) [35] S. Chen, F. Mallamace, C. Mou, M. Broccio, C. Cor-
[8] H. Stanley, S. Buldyrev, P. Kumar, F. Mallamace, saro,A.Faraone,andL.Liu,ProceedingsoftheNational
M. Mazza, K. Stokely, L. Xu, and G. Franzese, Journal Academy of Sciences 103, 12974 (2006)
of Non-Crystalline Solids(2010) [36] P. Kumar, S. Buldyrev, S. Becker, P. Poole, F. Starr,
[9] S. Buldyrev, P. Kumar, P. Debenedetti, P. Rossky, and andH.Stanley,ProceedingsoftheNationalAcademyof
H.Stanley,ProceedingsoftheNationalAcademyofSci- Sciences 104, 9575 (2007)
ences 104, 20177 (2007) [37] P. Kumar, G. Franzese, and H. Stanley, Physical review
[10] S. Han, M. Choi, P. Kumar, and H. Stanley, Nature letters 100, 105701 (2008)
Physics 6, 685 (2010) [38] P. Kumar and H. Stanley, J. Phys. Chem. A. DOI:
[11] P. Kumar, S. Han, and H. Stanley, Journal of Physics: 10.1021/jp2051867(2011)
Condensed Matter 21, 504108 (2009) [39] A. Schreiber, I. Ketelsen, and G. Findenegg, Physical
[12] S. Sastry, P. Debenedetti, F. Sciortino, and H. Stanley, Chemistry Chemical Physics 3, 1185 (2001)
Physical Review E 53, 6144 (1996) [40] P. Fedichev, L. Menshikov, G. Bordonskiy, and
[13] L.Rebelo,P.Debenedetti,andS.Sastry,TheJournalof A. Orlov, JETP Letters 94, 401 (2011), arxiv preprint
chemical physics 109, 626 (1998) arXiv:1104.1417
[14] P. Poole, F. Sciortino, U. Essmann, and H. Stanley, Na- [41] G. Bordonskiy and A. Orlov, Russian Journal of Con-
ture 360, 324 (1992) densed matter and interphase boundaries 13, 5 (2011)
[15] P.Poole,F.Sciortino,T.Grande,H.Stanley,andC.An- [42] J. Bernal and R. Fowler, J. chem. Phys 1, 515 (1933)
gell, Physical review letters 73, 1632 (1994) [43] P. Fedichev and L. Menshikov, eprint arXiv:
[16] L. Xu, P. Kumar, S. Buldyrev, S. Chen, P. Poole, 0808.0991(2008)
F.Sciortino,andH.Stanley,ProceedingsoftheNational [44] P. Fedichev and L. Menshikov, Arxiv preprint cond-
Academy of Sciences of the United States of America mat/0601129(2006)
102, 16558 (2005) [45] L.MenshikovandP.Fedichev,RussianJournalofPhys-
[17] C. Bertrand and M. Anisimov, The Journal of Physical ical Chemistry A 85, 906 (2011)
Chemistry B(2011) [46] Stogrin,IEEETrans.MicrowaveTheoryTech.MIT-19,
[18] H. Stanley, C. Angell, U. Essmann, M. Hemmati, 733 (1971)
P.Poole,andF.Sciortino,PhysicaA:StatisticalMechan- [47] H.Liebe,G.Hufford,andT.Manabe,InternationalJour-
icsanditsApplications205,122(1994),ISSN0378-4371 nal of Infrared and Millimeter Waves 12, 659 (1991),
[19] H. Tanaka, The Journal of chemical physics 105, 5099 ISSN 0195-9271
(1996) [48] J. Hasted, D. Ritson, and C. Collie, The Journal of
[20] S. Harrington, P. Poole, F. Sciortino, and H. Stanley, Chemical Physics 16, 1 (1948)
Journal of Chemical Physics 107, 7443 (1997) [49] P.Poole,T.Grande,F.Sciortino,H.Stanley,andC.An-
[21] M.Yamada,S.Mossa,H.Stanley,andF.Sciortino,Phys- gell, Computational materials science 4, 373 (1995)
ical review letters 88, 195701 (2002) [50] E.Ponyatovskii,V.Sinitsyn,andT.Pozdnyakova,JETP
[22] P. Jedlovszky and R. Vallauri, The Journal of chemical Lett 60, 360 (1994)
physics 122, 081101 (2005) [51] S.Myneni,Y.Luo,L.Näslund,M.Cavalleri,L.Ojamäe,
[23] D. Paschek, Physical review letters 94, 217802 (2005) H. Ogasawara, A. Pelmenschikov, P. Wernet, P. Väter-
[24] D.Paschek,A.Rüppert,andA.Geiger,ChemPhysChem lein,C.Heske,etal.,JournalofPhysics: CondensedMat-
9, 2737 (2008) ter 14, L213 (2002)
[25] Y. Liu, A. Panagiotopoulos, and P. Debenedetti, The [52] P. Wernet, D. Nordlund, U. Bergmann, M. Cavalleri,
Journal of Chemical Physics 131, 104508 (2009) M. Odelius, H. Ogasawara, L. Näslund, T. Hirsch,
[26] J. Abascal and C. Vega, J. Chem. Phys 133, 234502 L. Ojamäe, P. Glatzel, et al., Science 304, 995 (2004)
(2010) [53] T. Tokushima, Y. Harada, O. Takahashi, Y. Senba,
[27] T. Truskett, P. Debenedetti, S. Sastry, and S. Torquato, H.Ohashi,L.Pettersson,A.Nilsson,andS.Shin,Chem-
The Journal of chemical physics 111, 2647 (1999) ical Physics Letters 460, 387 (2008)
[28] T. Truskett and K. Dill, The Journal of Physical Chem- [54] C. Huang, K. Wikfeldt, T. Tokushima, D. Nordlund,
istry B 106, 11829 (2002) Y. Harada, U. Bergmann, M. Niebuhr, T. Weiss,
[29] H. Lee and R. Swendsen, Physical Review B 64, 214102 Y.Horikawa,M.Leetmaa,et al.,ProceedingsoftheNa-
(2001) tional Academy of Sciences 106, 15214 (2009)
6
[55] N. English and J. Tse, Physical Review Letters 106, [64] S. Jackson and R. Whitworth, The Journal of Physical
37801 (2011) Chemistry B 101, 6177 (1997), ISSN 1520-6106
[56] H.Frohlich,Theoryofdielectrics: dielectricconstantand [65] S. Singer, J. Kuo, T. Hirsch, C. Knight, L. Ojamae, and
dielectric loss (Clarendon press Oxford, 1949) M.Klein,Physicalreviewletters94,135701(2005),ISSN
[57] M. Krivoglaz and A. Smirnov, Teoriya uporyadochiv. 1079-7114
splavov (Theory of ordering alloys) (Fizmatgiz Moscow, [66] C. Knight, S. Singer, J. Kuo, T. Hirsch, L. Ojamäe, and
1958) M. Klein, Physical Review E 73, 056113 (2006)
[58] E. Moore and V. Molinero, The Journal of chemical [67] G.BarkemaandJ.DeBoer,JournalofChemicalPhysics
physics 132, 244504 (2010) 99, 2059 (1993)
[59] E.MooreandV.Molinero,Nature479,506(2011),arxiv [68] F. Stillinger, Philosophical transactions of the Royal So-
preprint arXiv:1107.1622 ciety of London. Series B, Biological sciences 278, 97
[60] L. Onsager, Journal of the American Chemical Society (1977)
58, 1486 (1936) [69] X. Su, L. Lianos, Y. Shen, and G. Somorjai, Physical
[61] M.BlackmanandN.Lisgarten,ProceedingsoftheRoyal Review Letters 80, 1533 (1998), ISSN 1079-7114
Society of London. Series A. Mathematical and Physical [70] M. Iedema, M. Dresser, D. Doering, J. Rowland,
Sciences 239, 93 (1957) W. Hess, A. Tsekouras, and J. Cowin, J. Phys. Chem.
[62] J. Finney, A. Hallbrucker, I. Kohl, A. Soper, and B 102, 9203 (1998)
D. Bowron, Physical review letters 88, 225503 (2002)
[63] S. Kawada, J. Phys. Soc. Japan. 32, 1442 (1972)
