We improve a lattice model of water introduced by Sastry, Debenedetti,
Sciortino, and Stanley to give insight on experimental thermodynamic anomalies
in supercooled phase, taking into account the correlations between
intra-molecular orientational degrees of freedom. The original Sastry et al.
model including energetic, entropic and volumic effect of the
orientation-dependent hydrogen bonds (HBs), captures qualitatively the
experimental water behavior, but it ignores the geometrical correlation between
HBs. Our mean-field calculation shows that adding these correlations gives a
more water-like phase diagram than previously shown, with the appearance of a
solid phase and first-order liquid-solid and gas-solid phase transitions.
Further investigation is necessary to be able to use this model to characterize
the thermodynamic properties of the supercooled region.Comment: 7 pages latex, 3 figures EP

Franzese, G.

Stanley, H. E.

Yamada, M.

English

arXiv

Giancarlo Franzese

Crossref

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0 Hydrogen-Bonded Liquids: Effects of
Correlations of Orientational Degrees of
Freedom
Giancarlo Franzese∗, Masako Yamada∗† and H. Eugene Stanley∗.
∗Center for Polymer Studies and Department of Physics, and †Center for Computational
Science Boston University, Boston MA 02215, USA.
Abstract.
We improve a lattice model of water introduced by Sastry, Debenedetti, Sciortino,
and Stanley to give insight on experimental thermodynamic anomalies in supercooled
phase, taking into account the correlations between intra-molecular orientational de-
grees of freedom. The original Sastry et al. model including energetic, entropic and
volumic effect of the orientation-dependent hydrogen bonds (HBs), captures quali-
tatively the experimental water behavior, but it ignores the geometrical correlation
between HBs. Our mean-field calculation shows that adding these correlations gives a
more water-like phase diagram than previously shown, with the appearance of a solid
phase and first-order liquid-solid and gas-solid phase transitions. Further investigation
is necessary to be able to use this model to characterize the thermodynamic properties
of the supercooled region.
INTRODUCTION
Water has a temperature density maximum at 4 oC at ambient pressure, above
the melting line. This is perhaps the most well-known anomaly of the many ob-
served in liquid water. In the metastable, supercooled region of water, the anoma-
lies are especially pronounced. The absolute magnitude of thermodynamic response
functions such as isothermal compressibility [1], thermal expansion coefficient [2],
and isobaric heat capacity [3] are all known to increase dramatically as the tempera-
ture is lowered. The quest for an explanation has led to extensive experimental, nu-
merical, and theoretical investigations of the properties of supercooled water [4–14]
However, in spite of many studies, a final answer has yet to be reached.
Three major theories have emerged from existing evidence. They are known
as the stability-limit conjecture [15], the two critical-point model [16], and the
singularity-free scenario [17–19] (Fig. 1). i) The stability-limit conjecture assumes
that the superheated liquid-to-gas spinodal and the supercooled liquid-to-solid spin-
odal in the pressure-temperature (P, T ) phase diagram are connected at negative
a) b) c)
L L L
G G G
C’ C C
P P P
T T T
HDL
LDL
TMD
> > >
> > >
FIGURE 1. Three interpretations of the supercooled water phase diagram: a) stability limit
conjecture, b) two critical-points model, c) singularity-free scenario. A dashed line denotes a
spinodal, a bold line denotes a transition line, and a dotted line denotes a TMD line. The
solid-liquid transition line is not shown; C is the known critical point; C′ is the hypothetical
second critical point.
P . ii) The two critical-point scenario does not assume such a retracing spinodal.
Instead, it attributes the anomalies of water to a second critical-point in the su-
percooled region of the phase diagram. This critical-point comes at the end of a
phase-transition line that separates two metastable liquid phases, a low-density liq-
uid (LDL), and a high-density liquid (HDL). iii) The singularity-free scenario does
not attribute the anomalies in water to a transition in the supercooled region or a
retracing spinodal, but to a line of temperatures of maximum density (TMD) in the
stable liquid phase and in the super-heated liquid phase with positive (∂P/∂T )TMD
(slope) in the P, T plane at high P and negative slope at lower P (retracing TMD).
The aforementioned response functions are not considered to diverge, but merely to
have maxima. Note that even in the two critical points scenario the TMD is retrac-
ing. There are theoretical calculations that show that the apparent gap between
these different scenarios can be reconciled without inconsistencies [20,21,13]. Thus,
the differences between the different scenarios may be much more subtle than ex-
pected. Authoritative experiments have not been forthcoming due to the difficulty
of probing metastable water without encountering inevitable crystallization [7].
It is worth noting that these so-called anomalies are not unique to water. Al-
though water exhibits anomalous properties relative to the liquid norm, namely
argon-like liquids, its unusual characteristics are shared by other fluids such as
SiO2 [22] and GeO2 [23]. What differentiates these anomalous liquids from nor-
mal liquids is that they form bonded networks that are dependent on the relative
orientation of the molecules.
The effect of orientation-dependent, hydrogen-bonded networks on critical behav-
ior has already been recognized. However, the effect of the intra-molecular correla-
tion between the orientational degrees of freedom has not be explored thoroughly.
For example, an open problem is how the intensity of orientational correlations can
change the phase diagram. This is where our focus lies.
THE MODEL
Our Hamiltonian can be expressed as:
H = −ǫ
∑
<i,j>
ninj − µ
∑
i
ni − J
∑
<i ,j>
ninj δσij σji − Jσ
∑
i
ni
∑
(k ,l)i
δσikσil . (1)
The first three terms constitute the original Hamiltonian [18,19]. The first two
terms in the Hamiltonian are the standard lattice gas Hamiltonian where ni = 0, 1
indicates whether site i is occupied by a molecule, ǫ is the attractive energy between
molecules, µ is the chemical potential and the symbol
∑
<i,j> means that the sum
is on all nearest neighbor (NN) lattice sites.
The third term in Eq.(1) describes the interaction of orientational degrees of
freedom (arms) between neighboring molecules, and accounts for hydrogen bonding
within the system. It is introduced in the following way: Each molecule has a max-
imum of γ neighbors; Therefore, it has γ arms that can potentially form HBs with
the respective arms of its neighbors. Each of the arms can assume q orientational
states, so the total number of orientational states for a molecule is qγ. The Potts
variable σij = 1, 2, . . . , q [24] specifies the orientational state of the arm of molecule
i that points toward nearest-neighbor molecule j. When two neighboring sites are
occupied (ni = nj = 1) and the orientational states of the two reciprocally pointing
arms match (σij = σji), a HB forms (δσijσji = 1 where δab = 1 if a = b, otherwise
it is δab = 0). Otherwise is ninjδσijσji = 0. This leads to an energy gain J per HB.
This third term is also summed over all NN sites.
The fourth term in Eq.(1) comes from our generalization, and takes into con-
sideration the interaction of orientational degrees of freedom within each molecule.
For an occupied site ni = 1, there is an energy gain Jσ if the orientations of two
arms, k and l, are not independent of each other (for the sake of simplicity, they
have to be in the same state in this model). If Jσ = 0 we recover the original
Sastry et al. Hamiltonian [18,19]. This term is summed over all sites, and over all
intra-molecular permutations of arms within each molecule.
As in the Sastry et al. model, in our generalization the creation of a HB not only
contributes to an energy factor, but also leads to an increase in volume:
V = Vo + dV NHB (2)
where Vo denotes the volume of the system without HBs, dV denotes the increase
in volume per HB, and NHB denotes the total number of HBs in the system. The
original model takes into account main ingredients: i) The increase of volume upon
HB formation (Eq.2); ii) The decrease of entropy upon HB formation due to the
decrease of orientational states available to hydrogen bonded molecules. To these
two features we add a third ingredient that is: iii) The correlation between the
orientations of different arms of the same molecules.
THE MEAN-FIELD CALCULATION
From the above explanations, we can see that there are two order parameters
in this system, a lattice gas order parameter m ∈ [−1, 1], and an orientational
order parameter s ∈ [0, 1]. The order parameter m indicates the proportion of
occupied sites and is related to the fraction of occupied sites (number of molecules
normalized to the total number of sites N) n =
∑
i ni/N but it does not give the
true density of the system ρ = nN/V where V is given by Eq.(2), affected by the
number of HBs. The orientational order parameter s is required to find the true
density of the system.
The orientational order parameter s is a measure of how much the preferred
orientational state dominates the system. When s = 0, all of the arms of occupied
sites assume a random orientational state from among the q possible states. No
orientational state is favored and network-formation is not encouraged. When
s = 1, all of the arms assume the same orientational state. Thus, hydrogen-bonded
network formation is heavily favored, provided sufficient number density. This
order parameter is analogous to the magnetization of an Ising system, where there
is no preferred orientational state initially, but where one state is spontaneously
“chosen” due to random fluctuations (symmetry breaking).
In mean field (MF) approximation the fraction of occupied sites n is supposed
to be a linear function of the lattice gas order parameter m as n = (1+m)/2 and,
analogously, the fraction of Potts variables in a given state (supposing a symmetry
breaking) is y = [1 + (q − 1)s]/q as function of the Potts MF order parameter s.
To find the equilibrium points we consider the MF molar Gibbs free energy per
site g = u− Ts+ Pv ≡ µ where
u = −
γ
2
[
ǫn+
(
Jn +
γ − 1
2
γJσ
)
(
y2 +
(1− y)2
q − 1
)]
(3)
is the molar energy per site,
s = −kB
[
lnn+
1− n
n
ln(1− n) + γ
(
y ln y + (1− y) ln
1− y
q − 1
)]
(4)
is the molar entropy per site (kB is the Boltzmann constant) calculated considering
that any molecule can have qγ states,
v =
v0
n
+
γ
2
nvHB
(
y2 +
(1− y)2
q − 1
)
(5)
is the molar volume per site (v ≡ 1/ρ) with v0 = V0/N and vHB = dV/N .
Using temperature and pressure as input variables, g is minimized with respect
to the order parameters. The resulting values for m and s are directly related to the
number density and hydrogen-bonding probability of that state point. Together,
these can provide the true density, which is needed to map out the phase diagram.
Previous two-dimensional mean-field calculations executed by Sastry et al. using
Eq.(1) without the fourth term, have shown that an orientationally-dependent,
network-bonded liquid without correlations between the orientational degrees of
freedom exhibits phase behavior as described by the singularity-free scenario [18,19].
(Fig.1c). When the fourth term of Eq.(1) is included an additional water-like
features appear in the phase diagram as shown in Fig.2. Namely, the MF calculation
recovers a new full-bonded (s = 1) phase that represents the solid phase. In
particular, we find a positively sloped first-order solid-gas transition line and a
negatively sloped first-order solid-liquid transition line ending in a tricritical point.
Furthermore it is possible to recover the same behavior for the liquid-gas spinodal
line (not shown in Fig.2). But, in the limit of this approach, is not possible to see for
each pressure a single point of maximum density (the TMD line), but only a range of
temperatures in which the density (molar volume) has a flat maximum (minimum).
For each pressure the lower temperature of this range is indistinguishable on the
scale of Fig.2 to the solid-liquid transition line, while the higher temperature is
shown in Fig.2 and has a positive slope. This result is not very different from
the MF result found for the Sastry et al. Hamiltonian with the assumption of no
correlation between orientational degrees of freedom, shown in Fig.5 of Ref. [19],
where the higher is the pressure the flatter is the minimum in the molar volume.
The only difference is that in our case a flat minimum is hardly distinguished by
a range of minima. Therefore this model in this MF approach leads to the same
singularity-free scenario for the density anomalies of the water already found in the
original model of Sastry et al. [18,19].
It is to be noted here that in our approach we are ignoring the fluctuations
and supposing that the Potts (orientational) symmetry breaking always occurs,
therefore we cannot have a new transition line ending in a new critical point.
The inclusion of fluctuations could be the essential feature to recover the second
critical point in the metastable supercooled liquid phase. Furthermore at negative
pressure and T = 0 a spurious state appears, due to the strong assumption (the
Potts symmetry breaking) that we made and to the fact that upon stretching the
system can minimize the Gibbs free energy forming HBs in the gas phase.
CONCLUSIONS AND OPEN ISSUES
By taking a simple model of water that leads to the singularity-free scenario
[18,19] as interpretation of the anomalous water behavior and adding a term that
correlates the intra-molecular orientational degrees of freedom, we recover the solid
phase, the liquid-solid and the gas-solid transition lines. Furthermore the TMD
“region” is above the melting line. All those are characteristic features of water
not included in the seminal Sastry et al. model [18,19]. This can be considered a
more realistic extension of the previously mapped singularity-free phase diagram,
which has no such transitions. However, these are stable transitions which by
themselves cannot verify or negate the existence of singularities in the metastable
a) b)
FIGURE 2. MF phase diagram for the model in Eqs.(1,2) with J/ǫ = 0.5, X/ǫ = 0.01,
vHB/v0 = 0.25, γ = 4, q = 10. a) The P, T projection: Lines represent phase transitions
with negatively sloped solid-liquid transition and liquid-gas transition ending in a critical point;
Stars represent the limits of the TMD region. b) The ρ, T projection for isobars ranging from
Pv0/ǫ = −.0007 (bottom) to Pv0/ǫ = 0.36 (up) with increment of ∆Pv0/ǫ = 0.03. Note that for
Pv0/ǫ = −.0007 there is a solid-gas transition, that is indistinguishable in a).
region, but they make the phase diagram more water-like.
So far our investigation is done only adding correlations between the orientational
degrees of freedom in the limit of no fluctuations. In order to understand the
critical phenomena of water in the supercooled region, thermal fluctuations cannot
be ignored. An inherent shortcoming of MF calculations is that such fluctuations
are blotted out by strong approximations. One way of including such fluctuations
into the analysis is to consider them into the MF calculations in a perturbative
manner, for instance by taking into account the local fluctuations of density [21].
Another way is to conduct numerical simulations [12]. We will now pursue these
two new paths to see what effect fluctuations have on the phase behavior of this
system, particularly in the metastable region.
ACKNOWLEDGMENTS
We would like to thank A. Scala for many enlightening discussions. G.F. is par-
tially supported by CNR grant N.203.15.8/204.4607 (Italy), M.Y. is supported by
a National Science Foundation traineeship administered by the Center for Com-
putational Science at Boston University, and the Center for Polymer Studies is
supported by the National Science Foundation.
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