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arXiv:0903.3914v1 [cond-mat.soft] 23 Mar 2009

Effect of concentration on the thermodynamics of sodium chloride aqueous solutions

in the supercooled regime

D. Corradini, P. Gallo∗and M. Rovere

Dipartimento di Fisica, Universit` a “Roma Tre”

Via della Vasca Navale 84, I-00146 Roma, Italy

Molecular Dynamics simulations are performed on two sodium chloride solutions in TIP4P water

with concentrations c = 1.36mol/kg and c = 2.10mol/kg upon supercooling. The isotherms and

isochores planes are calculated. The temperature of maximum density line and the limit of mechani-

cal stability line are obtained from the analysis of the thermodynamic planes. The comparison of the

results shows that for densities well above the limit of mechanical stability, the isotherms and iso-

chores of the sodium chloride aqueous solution shift to lower pressures upon increasing concentration

while the limit of mechanical stability is very similar to that of bulk water for both concentrations.

We also find that the temperature of maximum density line shifts to lower pressures and temper-

atures upon increasing concentration. Indications of the presence of a liquid-liquid coexistence are

found for both concentrations.

PACS numbers: 65.20.Jk,De,64.60.My

I. INTRODUCTION

The properties of aqueous ionic solutions besides being

of undoubtful importance in chemical physics1and elec-

trochemistry2, are relevant in many other fields of science

including biology and biophysics3, geophysics4, and even

atmospheric modeling.5In the supercooled region, ther-

modynamic properties of solutions are also of interest for

the cryopreservation of organs and food.6,7,8From a more

fundamental point of view an improved understanding of

the thermodynamics of these systems upon supercooling,

can help to shed light on the open questions on bulk liq-

uid water.9

It is well known that water presents, in the supercooled

region, peculiar thermodynamic behavior.10,11,12,13,14In

particular, the most striking effects are the existence of

a temperature of maximum density (TMD) line and the

divergence of the isothermal compressibility KT, of the

isobaric specific heat cP and of the coefficient of thermal

expansion αP. The origin of this anomalous behavior is

still a matter of large interest and debate in the liter-

ature.11Several theoretical15,16,17,18,19,20,21,22and com-

puter simulation23,24,25,26,27,28,29,30,31papers have shown

the presence in the supercooled region of water of a

liquid-liquid (LL) critical point. Experimental signatures

of this critical point have been also found.32The sec-

ond critical point of water would be the end point of

the coexistence line between a low density liquid (LDL)

and a high density liquid (HDL). In this framework, the

anomalous properties of water arise as a consequence of

the presence of the LL critical point. Furthermore in this

picture, the limit of mechanical stability (LMS) is non-

reentrant and the TMD line is knee-shaped and avoids

∗Author to whom correspondence should be addressed; e-mail: gal-

lop@fis.uniroma3.it

to cross the LMS line.

Aqueous ionic solutions have been extensively stud-

iedatambient temperature.

properties arethe main

with particular emphasis on the hydration struc-

ture.33,34,35,36,37,38,39,40,41,42,43,44,45Many studies in the

supercooled regime deal with the glass transition phe-

nomenon (see Ref. 1 and references therein) while the

detailed comparison of the thermodynamic behavior of

the aqueous solutions with respect to bulk water in the

mild supercooled regime still lacks a thorough investi-

gation. Calorimetric experiments have shown that from

low to moderate concentration of ions several thermody-

namic properties of aqueous solutions are dominated by

the solvent.5,46

In this paper, we present a Molecular Dynamics (MD)

simulation study of the thermodynamics of two sodium

chloride aqueous solutions, in the following denoted also

as NaCl(aq), in the supercooled regime. This work is an

extension of a previous study performed on bulk water

and on a NaCl(aq) solution with low salt concentration.47

The concentrations of salt in the solutions studied

in the present work are c = 1.36mol/kg and c =

2.10mol/kg. For both systems we study the isotherms

in the P −ρ plane and the isochores in the P −T plane.

The analysis of those thermodynamic planes leads to the

determination of the LMS and TMD lines. Moreover we

present the trend of the potential energy as a function

of density, at a low temperature. We will compare the

results of the present simulations with results on bulk

water and c = 0.67mol/kg NaCl(aq) studied in our pre-

vious work.47We also perform a comparison of the results

with what found for water confined in a hydrophobic en-

vironment of soft spheres.48

The paper is organized as follows. In Sec. II we explain

the details of the model and the computer simulation

setup. In Sec. III we show and discuss the thermody-

namic behavior. Conclusions are drawn in Sec. IV.

Their

of

structural

most papers,focus

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2

200

250

300

350

T (K)

400

450500

-100

0

100

200

300

400

500

P (MPa)

Pettitt-Rossky c = 0.67

Jensen-Jorgensen c = 0.67

Pettitt-Rossky c = 1.36

Jensen-Jorgensen c = 1.36

ρ = 1.025

FIG. 1: Isochores for c = 0.67mol/kg NaCl(aq)47and for c =

1.36mol/kg NaCl(aq) for ρ = 1.025 g/cm3for two different

force fields50,57(color online).

II. MODEL AND SIMULATION DETAILS

Two aqueous sodium chloride solutions with concen-

trations, given in moles of solute per mass of solvent, c =

1.36mol/kg and c = 2.10mol/kg are simulated by means

of MD technique. In the case of the c = 1.36mol/kg so-

lution the system is composed by 244 water molecules,

6 Na+ions and 6 Cl−ions, while in the case of the

c = 2.10mol/kg solution, it is composed by 238 water

molecules, 9 Na+ions and 9 Cl−ions.

The particles interact via the sum of coulombic and

Lennard-Jones (LJ) potentials. The analytical expres-

sion of the interaction potential is given by

Uij(r) =qiqj

r

+ 4ǫij

??σij

r

?12

−

?σij

r

?6?

(1)

where q is the electric charge and ǫijand σij are LJ pa-

rameters. Water molecules are modeled employing the

TIP4P potential.49Ion-ion and ion-water LJ parame-

ters are derived from Pettit and Rossky50parameters for

the Huggins-Mayer potential, via the reparametrization

made by Koneshan and Rasaiah35for LJ potential. The

ion-water and ion-ion LJ interaction parameters are sum-

marized in Table I.

Periodic boundary conditions are applied. The cutoff

radius is fixed at 9.0˚ A. Usually cutoff radius is fixed in

simulations between 8 and 10˚ A51. Long range electro-

static interactions are taken into account by the Ewald

summation method with convergence parameter α set to

6.4/L, where L is the edge of the cubic simulation box.

200

250

300

350

400

450500 550

T (K)

-200

-100

0

100

200

P (MPa)

Ntot = 256

Ntot = 512

Ntot = 1024

c = 2.10

c = 1.36

FIG. 2: Isochores for c = 1.36mol/kg NaCl(aq) and for c =

2.10mol/kg NaCl(aq) for ρ = 0.98 g/cm3for three different

sistem sizes (color online).

TABLE I: Ion-water and ion-ion LJ interaction parameters.

Atom pair

Na-O

Na-H

Cl-O

Cl-H

Na-Na

Cl-Cl

Na-Cl

ǫ(kJ/mol)

0.56014

0.56014

1.50575

1.50575

0.11913

0.97906

0.35260

σ(˚ A)

2.720

1.310

3.550

2.140

2.443

3.487

2.796

The systems are equilibrated by controlling the tem-

perature with the Berendsen thermostat.52Production

runs are done in the NV E ensemble. The integration

timestep used is 1 fs.

For both systems we studied the densities ρ

1.125,1.1,1.05,1.025,0.98,0.95,0.90,0.87,0.85,0.80g/cm3.

For each density, a starting configuration is produced

distributing the particles on a face centered cubic lattice,

with random orientation of water molecules. The crystal

is then melted at T = 500 K and the temperature is

stepwise reduced during the equilibration runs.

lowest temperature investigated is T = 190 K. Equilibra-

tion runs become very long for the lowest temperatures

investigated. Each equilibration run is followed by a

production run in which the thermodynamic averages

are calculated. Production runs are always done with

=

The

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0.80.9

11.1

ρ (g/cm3)

-300

-200

-100

0

100

200

P (MPa)

T = 300 K

T = 280 K

T = 260 K

T = 250 K

T = 240 K

T = 230 K

T = 220 K

T = 210 K

LMS

c = 1.36 mol/kg

FIG. 3: Isotherms in the range 210K ≤ T ≤ 300K and LMS

line of c = 1.36mol/kg NaCl(aq) in the P − ρ plane (color

online).

the same length of the equilibration runs. The longest

equilibration and production runs last up to 10 ns each.

The simulations are carried out using the DL POLY

package.53The pressures extracted are calculated with

the virial equation54.

The choice of the force field is very important in the

case of ionic aqueous solutions41,55since for example in

KCl recent studies have evidenciated possible problems

that are water model independent56. However the NaCl

behavior seems to show an weaker dependence on the

specific force field since it shows a lower tendency to form

clusters56. For the c = 2.10mol/kg solution and ρ = 1.1

g/cm3at T = 300 K we have an internal energy value

of −62.81 kJ/mol. This value can be compared with

a simular value of −69.57 kJ/mol obtained for a c =

2.35mol/kg solution and ρ = 1.093g/cm3at T = 300 K

for a ionic potential with SPC flexible water potential58.

In order to stringently test the robustness of our potential

we have run simulations along a isochore with a recent

ionic potential by Jensen and Jorgensen57tailored for

TIP4P water. In Fig. 1 we show, for the two different

concentrations, the isochore ρ = 1.025 g/cm3calculated

with both potentials. We can see that the two potentials

produce similar results.

We have also conducted a test to verify that our data

do not depend significatively on the size of the box. Re-

sults are reported in Fig. 2. For c = 1.36mol/kg we

compare the ρ = 0.98 g/cm3isochore calculated for

244 water molecules and 6 ion pairs, and for 488 wa-

ter molecules and 12 ion pairs. The simulation box of

these systems is L = 20.037, 25.2468˚ Arespectively. For

0.8 0.9

11.1

ρ (g/cm3)

-300

-200

-100

0

100

200

300

P (MPa)

T = 400 K

T = 350 K

T = 300 K

T = 280 K

T = 260 K

T = 250 K

T = 240 K

T = 230 K

T = 220 K

T = 210 K

LMS

c = 2.10 mol/kg

FIG. 4: Isotherms in the range 210K ≤ T ≤ 400K and LMS

line of c = 2.10mol/kg NaCl(aq) in the P − ρ plane (color

online).

c = 2.10mol/kg we compare the ρ = 0.98 g/cm3isochore

calculated for 238 water molecules and 9 ion pairs, for

476 water molecules and 18 ion pairs and for 952 water

molecules and 36 ion pairs. The simulation box of these

systems is L = 20.132, 25.365, 31.958˚ Arespectively. We

note that the curves corresponding to the same concen-

trations are very similar and that their minimum does

not show any significant shift.

III. THERMODYNAMIC RESULTS

The simulated thermodynamic state points have been

reported in the P −ρ (isotherms) plane and in the P −T

(isochores) plane. The analysis of those planes allows the

determination of the LMS line and TMD line, respec-

tively. Both curves can be derived using thermodynamic

relations.

By considering the isothermal compressibility

KT= −1

V

?∂V

∂P

?

T

=1

ρ

?∂ρ

∂P

?

T

(2)

the LMS line is defined by the locus of the points for

which KTdiverges. The line that joins the minima of the

isotherms corresponds to the LMS line. The TMD line

is defined as the locus of the points where the coefficient

of thermal expansion αP is zero.

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200

250

300

350

T (K)

400

450 500550

-400

-200

0

200

400

600

P (MPa)

c = 1.36 mol/kg

ρ = 1.125

ρ = 1.1

ρ = 1.05

ρ = 1.025

ρ = 0.98

ρ = 0.95

ρ = 0.90

LMS

TMD

FIG. 5: Isochores in the range 0.90g/cm3≤ ρ ≤ 1.125g/cm3,

TMD and LMS lines of c = 1.36mol/kg NaCl(aq) in the P−T

plane (color online).

αP=1

V

?∂V

∂T

?

P

= −1

ρ

?∂ρ

∂T

?

P

= KT

?∂P

∂T

?

ρ

(3)

Therefore the line joining the minima of the isochores

yields the TMD line.

In Fig. 3 and Fig. 4 we report the isotherms of the two

solutions in the P −ρ plane as given by our simulations.

In both cases we display only the curves that show min-

ima and thus contribute to the calculation of the LMS

line.

Fig. 3 refers to the c = 1.36mol/kg solution.

show the isotherms in the range 210K ≤ T ≤ 300K

and the corresponding LMS line.

highest temperature isotherm which shows a minimum.

The LMS line starts at ρ = 0.87g/cm3at T = 300 K

and shifts to ρ = 0.90g/cm3for all the lower temper-

ature curves. The lowest temperature isotherms, T =

220 K and T = 210 K, show inflections that cross the

higher temperature isotherms for densities in the range

0.98g/cm3≤ ρ ≤ 1.05g/cm3.

Fig. 4 refers to the c = 2.10mol/kg solution.

report the isotherms in the range 210K ≤ T ≤ 400K

and the corresponding LMS line.

tion, minima of the isotherms can be found up to the

T = 400 K isotherm. The LMS line gradually shifts to-

ward higher densities upon decreasing the temperature.

In this case the only isotherm showing an inflection is

the one at T = 210 K. This inflection spans the density

range 1.025g/cm3≤ ρ ≤ 1.05g/cm3.

We

T = 300 K is the

We

At this concentra-

200

250

300

350

T (K)

400

450 500550

-400

-200

0

200

400

600

P (MPa)

c = 2.10 mol/kg

ρ =1.125

ρ =1.1

ρ =1.05

ρ =1.025

ρ =0.98

ρ =0.95

ρ =0.90

TMD

LMS

FIG. 6: Isochores in the range 0.90g/cm3≤ ρ ≤ 1.125g/cm3,

TMD and LMS lines of c = 2.10mol/kg NaCl(aq) in the P−T

plane (color online).

Upon comparing the isotherms planes of the two so-

lutions we note that, at high densities, the isotherms of

the higher concentration solution are shifted by about

50 MPa toward lower pressures, with respect to the

isotherms of the lower concentration solution.

shift decreases at densities close to the minima of the

isotherms and it almost disappears for very low densi-

ties. This behavior of the isotherms is analogous to what

found in the comparison between bulk water and the

c = 0.67mol/kg solution.47A similar pressure shift can

be seen also when comparing the c = 1.36mol/kg and the

c = 0.67mol/kg solutions (not shown). Therefore upon

increasing ions concentration the isotherms progressively

shift toward lower pressures.

For the c = 1.36mol/kg solution the LMS line in the

isotherms plane (Fig. 3) is monotonic as already found for

bulk water23,30,59, confined water48and c = 0.67mol/kg

NaCl(aq).47In the c = 2.10mol/kg solution (Fig. 4) it

does not decrease on going from the T = 230 K to the

T = 220 K isotherm.

An important feature of the isotherms planes of the two

solutions is the presence of inflections of the low tempera-

ture isotherms. It has been previously shown for bulk wa-

ter that those inflections in the isotherms are a signature

of the approach of the systems to liquid-liquid (LL) coex-

istence.23,30,59As already noted for the c = 0.67mol/kg

solution47, this behavior is maintained in the NaCl(aq).

Therefore we can infer that the HDL/LDL coexistence,

possibly terminating in a second critical point, is present

in the NaCl(aq) solutions, at least up to c = 2.10mol/kg

concentration. Nonetheless the shrinkage of the density

This

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220240

260

280300

T (K)

-300

-200

-100

0

100

P (MPa)

FIG. 7: Isochores in the temperature range 210K ≤ T ≤

300K, starting from the top, for bulk water (diamonds),

c = 0.67mol/kg (circles), c = 1.36mol/kg (squares) and

c = 2.10mol/kg (triangles) solutions at densities ρ = 1.05

(filled symbols) and 0.90 (unfilled symbols) g/cm3(color on-

line).

range of inflections in the isotherms seems to indicate a

gradual disappearance of the coexistence upon increasing

salt content.

These findings are consistent with what found by

Archer and Carter5in their experimental paper. They

found that the anomalous behavior of supercooled wa-

ter, and in particular the divergence of isobaric specific

heat and the existence of a TMD line are maintained in

NaCl(aq) up to concentrations of about 2mol/kg. Thus,

in the framework of the second critical point scenario,

it could be proposed that those anomalies are a conse-

quence of a second critical point in the NaCl(aq) system,

shifted toward lower pressures with respect to bulk water.

In Fig. 5 and Fig. 6 the isochores planes of the two

solutions are reported. The isochores are presented along

with the LMS lines and the TMD lines. For both systems

the isochores lying below the LMS line are not reported.

The minima are obtained by fitting the isochores with

fourth degree polynomial functions.

In Fig. 5 we show the isochores in the range

0.90g/cm3≤ ρ ≤ 1.125g/cm3, the LMS line and the

TMD line for the c = 1.36mol/kg solution. The range of

temperatures spanned is 210K ≤ T ≤ 500K. All the iso-

chores above ρ = 0.90g/cm3display a minimum, while

the ρ = 0.90g/cm3isochore is almost completely coinci-

dent with the LMS line. Such LMS line is entirely in the

region of negative pressure and it is nonre-entrant down

to the lowest temperature we simulated. This behavior

-300

-200

-100

0

100

200

P (MPa)

210240 270300

T (K)

-300

-200

-100

0

100

200

P (MPa)

210 240270300

T (K)

Bulk c = 0.67 mol/kg

c = 1.36 mol/kg c = 2.10 mol/kg

FIG. 8:

P − T

0.67mol/kg NaCl(aq)47(top right panel), c = 1.36mol/kg

NaCl(aq) (bottom left panel) and c = 2.10mol/kg NaCl(aq)

(bottom right panel), in the temperature range 210K ≤

T

≤

300K and, starting from the top, for densities

ρ = 1.05,1.00,0.98.0.95, 0.90,0.87,0.85g/cm3for bulk wa-

ter, and for densities ρ = 1.05,1.025,0.98.0.95, 0.90g/cm3

for NaCl(aq) solutions (also ρ = 0.87g/cm3only for the

c = 0.67mol/kg solution), (color online).

Isochores and LMS lines (open circles) in the

plane for bulk water47

(top left panel), c=

has been already found in bulk water23,24,30,59,60,61, con-

fined water48and c = 0.67mol/kg NaCl(aq).47

In Fig. 6 the isochores in the range 0.90g/cm3≤

ρ ≤ 1.125g/cm3, the LMS line and the TMD line for

the c = 2.10mol/kg solution are displayed.

this case the range of temperatures spanned is 210K ≤

T ≤ 500K. The isochores above the ρ = 0.95g/cm3

show a minimum. At this concentration some oscilla-

tions can be found in the LMS line at low temperatures.

This line approximately follows the trend found for the

ρ = 0.90g/cm3isochore.

The comparison of the two isochores planes shows that

also the isochores of the c = 2.10mol/kg solution are

shifted toward lower pressures by roughly 50 MPa, with

respect to the c = 1.36mol/kg solution, as already noted

for the isotherms of the systems. This pressure shift de-

creases at low densities and at ρ = 0.90g/cm3it almost

vanishes.

In order to have a direct comparison of the two aqueous

solutions studied here with both bulk water and the c =

0.67mol/kg solution, we report in Fig. 7 the isochores for

bulk water, c = 0.67mol/kg, c = 1.36mol/kg and c =

2.10mol/kg solutions in the temperature range 210K ≤

T ≤ 300K and for densities ρ = 0.90g/cm3and ρ =

Also in