A high pressure calorimetric experiment to validate the liquid-liquid critical point hypothesis in water

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by  Manuel I. Marques

A high pressure calorimetric experiment to validate the liquid-liquid critical point
hypothesis in water
∗
Manuel I. Marqu´es
Departamento de F´ısica de Materiales C-IV,
Universidad Aut´onoma de Madrid, 28049 Madrid, Spain
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An experimentalproposal totest theexistenceof a liquid-liquidcritical point in water, based on
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high pressure calorimetric measurements ,is presented on this paper. Considering the existence of
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an intramolecular correlation in the water molecule we show how the response of the specific heat
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at high pressure is different depending on the existence, or not, of the second critical point. If the

n liquid-liquidcritical point hypothesisistruetheremust beamaximuminthespecificheat at some
a temperature T >TH for any pressure P >Pc (being TH the homogeneous nucleation temperature
J
andPc thepressureofthesecondcriticalpoint). Thismaximumdoesnotappearforthesingularity
3 free scenario.
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] Liquid water exists in a metastable supercooled state related to the formation of the hydrogen bond network
t
f farbellowthemeltingtemperature. Anumberofthermo- does not diverge and the thermodynamic response func-
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dynamicresponsefunctions ofsupercooledwatersuchas tions present a simple (non-divergent) maximum at a
s
t. the isothermalcompressibilityand the constantpressure temperature Th(P). This temperature, Th(P), is always
a specific heat, show a power-law divergence at T=228K proportional to the absolute value of the interaction en-
m
and P=1atm [1, 2]. In order to explain these anomalies ergy due to the formationof a hydrogenbond at a given
- several theories have been proposed. pressure,Th(P)∝(J−(PδV))[5]. NotethatTh(P)∝J
d
n The liquid-liquid phase transition hypothesis [3] pro- forP =0andTh(P)=0forP =J/δV. Thebehaviorof
o poses the existence of a first order line of phase transi- this simple model is the one described by the singularity
c tions separating two liquid states of different densities: free scenario.
[
the highdensity liquid (HDL) and the low density liquid However by considering an internal correlationamong

1 (LDL) state. This line of phase transitions has a nega- the four hydrogen bonds formed by each single molecule
v tive slope in the P −T phase diagram and ends up in a the scenario changes from singularity- free to critical
6 critical point at Tc ∼ 200K and Pc ∼1.7kbars. Another point like [5]. This internal degree of correlation in the
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possible scenario is singularity-free [4]. In this proposal water molecules is tuned by a new intramolecular term
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1 the anomalies found in the experiments are not consid- with energy −Jσ added to the Hamiltonian of the liquid
0 ered to end up in real singularities but in non divergent [5]. If Jσ → ∞ the water molecule is always internally
7 maxima due to the anticorrelatedfluctuations of volume correlated and, for any value of the pressure, the ther-
0 and entropy. modynamic functions are always divergent at Th(P).
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t The direct observation of a possible first order liquid- The liquid-liquid critical point at Tc and Pc shows up
a
m liquid phase transitionline and a secondcriticalpoint in whenJσ isfiniteandsmallerthanJ. Inthiscase,thewa-
water has been hampered by the intervention of the ho- ter molecules are able to correlate themselves internally
-
d mogeneousnucleationprocesswhichtakesplaceathigher onlyifT <T∗,beingT∗ apressureindependenttempera-
n temperatures TH(P) than the ones corresponding to the ture proportionaltoJσ. ForP <Pc the watermolecules
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hypothesized coexistence line. In this paper we are go- are not internally correlatedat Th(P) (i.e. Th(P)>T∗),
c
ing to focus our attentionin the following question: Can the correlation length of the hydrogen bonds network is
:
v weexpectanydifference(attemperaturesT >TH(P))in unable to tend to infinity, and the response of the ther-
i
X thecalorimetricresponseofsupercooledwaterdepending modynamic functions is non-divergent.
r on the existence, or not, of a second critical point? However, as the pressure increases Th(P) decreases.
a In order to give a possible answer to this question we So, there is a criticalvalue ofthe pressureP =Pc where
aregoingto consideraliquid modelwherethe existence, the water molecules are internally correlated at Th(Pc)
or not, of a second liquid-liquid critical point is easily (i.e. Th(Pc) ≤ T∗), the correlation length of the hydro-
tunable. The model is described in detail in [5]. In the gens bond network is able to go to infinity, and the re-
liquid phase, each water molecule interacts with another sponse of the thermodynamic functions is divergent at
four water molecules forming hydrogen bonds with en- Th(Pc) = Tc. Of course, for any P > Pc, the four hy-
ergy −J. The formation of a hydrogen bond increases drogen bonds formed by any water molecule are always
the volume of the system by a certain amount δV. In internallyorderedatTh(P)<T∗ andtheresponseofthe
the simplest version of this model, there is no correla- thermodynamic functions are going to be the ones cor-
tion between the possible four hydrogen bonds formed responding to a first order phase transition between two
by a particular water molecule. The correlation length liquids with different density.
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Basedonthe model, we proposethe followinghypoth-
esis: If there is a second critical point at Pc and we cool
downour liquid deepinto the supercooledregionat con-
stant P > Pc, there is going to be internal correlation
inside the water molecule at a temperature T∗ >Th (i.e.
before homogeneous nucleation takes place). The fluc-
tuations on the energy of the system due to this intra-
molecularorderinggiverisetoamaximuminthespecific
heatwith the following characteristics:(i)It is locatedat
a temperature larger than the one corresponding to nu-
cleation (T∗ > Th), (ii) it is non-divergent (because it is
not due to the cooperative effect of many molecules but
to the internalorderingofeachmolecule separately)and
(iii) the position ofthe maximum is going to be pressure
independent (i.e. since Jσ is constant and does not de-
pend on pressure the value of T∗ should be also pressure
independent).
On the other hand if P < Pc and we cool down our
liquid, nucleation takes place before intra-molecular or-
dering (T∗ < Th) and no maximum appears in the spe-
cific heat (only the regular nucleation maximum reach-
able in numerical simulations but unreachable in a real
supercooled bulk water experiments). In the case of a
singularity free scenario where there is no critical point
(criticalpressure),thereisneveranyinternalorderingin
FIG. 1: Three possible scenarios depending on the value of
the water molecules and, independently of the pressure
Jσ. (a) IfJσ =0 thereis nointernal correlation in thewater
applied, no maximum in the specif heat is found.
moleculeforanypressure(T∗ =0)andthereisnolongrange
correlationintheformationofthehydrogenbondednetwork. In order to test this hypothesis we have simulated the
The transition is always singularity-free (dotted line). (b) liquid phase of the model described in Ref. [5] by com-
If Jσ < J and different from zero, there is long range cor- puting the total energy density of states [9]. We have
relation in the transition to a low density liquid only when chosenthe followingvalues for the parameters: J/ǫ=50
the transition temperature Th is smaller than the tempera- (being ǫ the value of the van der Waals interaction en-
tureT∗ correspondingtotheinternalcorrelation ofthewater
ergy which in our case, since we are only considering
molecules. In this particular case we obtain a critical point
the condensed liquid phase, does not play any role) and
andafirstorderphasetransitionforanypressureP >Pc and
foranytemperatureT <Tc ∼T∗(grayregionandcontinuous δV/vo = 50 (being vo the hard core volume of the wa-
line). (c) IfJσ is large enoughcompared toJ thereisalways ter molecule). We have performed two sets of calcula-
internal correlation in the water molecules for any pressure tions, one for a liquid with no liquid-liquid critical point
andthetransitiontoalowdensityliquidisfirstorderforany (Jσ/ǫ = 0) and another one with a liquid-liquid critical
pressure (gray region and continuousline).
point (Jσ/ǫ = 5). All other parameters are the same
than in Ref. [5]. The behavior of energy fluctuations
(constant pressure specific heat) has been calculated for
This mechanism is explained in detail on Fig.1. Note a system of 225 water molecules with periodic boundary
how, by tunning the intra-molecular term Jσ, we can conditions and for the following values of the pressure
easily change the physical mechanism of the simulated Pvo/ǫ = 0.54,0.74,0.84,0.89,0.94,0.955,0.97. Results
liquidformsingularityfree(Jσ =0)toliquid-liquidphase are shown in Fig.2
transition (06=Jσ <J). Fig.2 (a) are the results for the liquid with no-critical
In real water, this intra-molecular term in the water point. Inthiscasenosecondarymaximumisfoundatany
molecule could be related to the value of the H-O-H pressure. Fig.2 (b) are the results for the liquid with a
angle. Experiments show that the relative orientations criticalpointandacoexistencelinebetweenaLDLphase
of the arms in the water molecule are correlated, with and a HDL phase. Note how the specific heat, for pres-
o
the averageH-O-Hangle equalto 104.45 inthe isolated suresPvo/ǫ>0.89,hastwomaxima. Thecharacteristics
o o
molecule, 104.474 in the gasand 106 in the highT liq- ofthesecondarypeaksarethefollowing: theyarelocated
uid[6], suggestinganintramolecularinteractionbetween always at a temperature (T∗) larger than the nucleation
thearms. Thisinteractionmustbefinitebecausethean- temperature, they are non-divergent, and the tempera-
glechangeswithT,consistentwithab-initiocalculations ture (T∗) is pressure independent. These findings agree
[7] and molecular dynamics simulations [8]. with our hypothesis previously presented and suggest
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1 10
a) b)
8
6
C00,5 SMeacxoinmduarmy
Ce/
4
2
0 0
0 2 4 6 8 10 0 2 4 6 8 10
T/ε T/ε
T/ε
*
FIG. 2: Constant pressure specific heat vs. tem-
perature for different pressures Pvo/ǫ =0.54 (thiner
line),0.74,0.84,0.89,0.94,0.955,0.97 (thicker line) and for (a)
Jσ/ǫ=0(singularity free scenario) and (b) Jσ/ǫ=5 (liquid-
liquid critical point scenario). Note how we find secondary
maximaonlyin(b)andforPvo/ǫ>0.89, indicatingthatthe FIG.3: Schematicrepresentationoftheexperimentproposed
critical pressure of themodel must be close to Pvo/ǫ=0.89. totesttheexistenceofasecondliquid-liquidcritical pointat
The relative values of the maxima of the peaks are system- Pc ∼1.7 kbar and Tc ∼200K (black circle) . Constant pres-
size dependent expect for the secondary peaks indicated in
surespecificheatmeasurementsshouldbeperformedbycool-
(b). Absolute specific heat values are normalized to C0, the ing down water until reaching the homogeneous nucleation
valueof thesecondary maxima found in (b). temperature(squares). Forpressuressmallerthan∼1.7kbar
the specific heat should present the usual constant power-
law increase when cooling in the deeply supercooled region.
that there is a critical point at a pressure Pvo/ǫ ∼ 0.89 However, for larger pressures (in the range form 1.7kbar to
for liquid (b). 2kbar), there should be a previous non-divergent maximum
Thereisanimportantcommentaboutthevaluesofthe in T = T∗ and close to the temperature of 200K indepen-
dently of the pressure applied. This behavior for the specific
main maxima of the specific heat shown in Fig.2. Since
heat has been schematically represented on the figure (thick
we are dealing with an small system with finite size ef-
continuous lines). Dotted line is the proposed coexistence
fects the concrete value of the maxima are not represen- line.
tative. For example, for the isothermal compressibility,
it is found that maxima are proportional to the number
of molecules in the first order phase transition, and that lowing our hypothesis, if there is a liquid-liquid critical
they scale as a power of the number of molecules when point, calorimetric measurements should present a non-
the transitionis atthe criticalpoint(secondorderphase divergent maxima for values of the pressure P > Pc ∼
transition)[5]. Inany case,they arenotmeasurablein a 1.7kbar[11]atatemperatureT∗ abovethehomogeneous
real experiment due to the homogeneous crystallization nucleation temperature (which is located between 200K
of water. On the contrary,the secondarymaxima shown and 180K depending on the value of the pressure ap-
on Fig.2 (b) are clearly non-divergent because they are plied). The value of T∗ should be almost pressure inde-
size independent (not shown). pendent and close to the hypothesized critical tempera-
This non-divergent secondary peak found for P > Pc ture Tc ∼ 200K [11]. This secondary maximum should
could be measured in a real, high pressure, water ex- notshowupinotherquantitiessuchasthe compressibil-
periment, since it is located at temperatures above the ity because the intra-molecular term we are considering
nucleation temperature. The scheme of the concrete ex- forthe watermolecule doesnotimply any changeinvol-
periment to be performed is presented on Fig.3. Calori- ume. On the other hand, if the real scenario in water is
metric measurements on water should be performed for singularity-freethereshouldbenomaximainthespecific
pressures ranging from atmospheric pressure to approx- heat.
imately P = 2 kbar in the supercooled region. Until Toconclude,theliquid-liquidcriticalpointandtheex-
now,calorimetricmeasurementsinbulkwaterhavebeen istence of an internal correlation in the water molecule
mainly performed at low pressures [10] and no maxi- could be strongly supported by the existence of non-
mum has been found down to T = 240K. However, fol- divergentmaximainthewaterspecificheatathighpres-
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sures. Of course, an important question is: in case these [4] H. E. Stanley and J.Teixeira, J. Chem. Phys. 73, 3404
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E 67 011103 (2003).
secondcriticalpointinwater? Theanswerisno,because
[6] C.W.KernandM.KarpulsinWater: AComprehensive
there is also the possibility of the existence of a liquid-
Treatise, edited by F. Franks(Plenum Press, New York,
liquid critical point in water that is not related to an
1972), Vol.1 pp. 21-91; V. B. Hasled, ibid., pp. 255-309;
internal correlationinside the water molecule due to the K. Ichikawa et al. Mol. Phys. 73, 79 (1991).
existence of anintra-molecularterm in the Hamiltonian. [7] P. L.Silvestrelli and M. Parrinello, J. Chem.Phys. 111,
3572 (1999).
[8] P. A. Netz, F. Starr, M. C. Barbosa and H. E. Stanley,
Physica A 314, 470 (2002).
[9] F. Wang and D. P. Landau, Phys. Rev. E 64 056101
∗ (2001).
Electronic address: [email protected]
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