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Polyamorphism and polymorphism of a confined water monolayer: liquid-liquid critical point, liquid-crystal and crystal-crystal phase transitions
Valentino Bianco, Oriol Vilanova, Giancarlo Franzese Departament de F´ısica Fonamental, Universitat de Barcelona Diagonal 647, 08028 Barcelona, Spain
Water is an anomalous liquid because its properties are different from those of the majority of liquids. Here, we first review what is anomalous about water. Then we study a many-body model for a water monolayer confined between hydrophobic plates in order to answer fundamental questions related to the origin of its anomalies and to predict new testable futures. In particular, we study by Monte Carlo simulations the low temperature phase diagram of the model. By finite size scaling, we find a liquid-liquid first order phase transition ending in a critical point (LLCP) in the region in which bulk water would be supercooled. We show that the LLCP belongs to the universality class of the two-dimensional (2D) Ising model in the limit of infinite walls. Next, we study the limit of stability of the liquid phase with respect to the crystal phases. To this goal we modify the model in order to characterize the crystal formation and find that the model has a crystal-crystal phase transition and that the LLCP is stable with respect to the liquid-crystal phase transition depending on the relative strength of the three-body interaction with respect to the rest of many-body interactions.
1. An overview of water A mono-component substance is called anomalous if it behaves differently with respect to argon-like liquids.1 A few liquids are found to show anoma- 2–4 5,6 7 lous properties: silica (SIO2), silicon (Si), selenium (Se), phosphorus (P),8–10 among others. However, the most famous example is water.11 Water is undoubtedly the most important liquid for life as we know it. More than 60% of the human body is composed by water. Almost all biological processes need water to work correctly. Water is largely used in modern technologies and industries. Climate changes are strikingly related to water cycling. Among all substances in nature, it is the only one occurring in solid, liquid and vapor phases at ambient conditions. Despite the fact that liquid water is so common in nature, it has many September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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anomalous properties–more than sixty–with respect to other liquids. A fa- mous example is its maximum of density at approximately 4◦C, leading to a crystal that is less dense than the liquid. In the following we will review some anomalies properties of water. These anomalies are stronger in the metastable liquid phase below the melt- ing temperature. Water is a strongly metastable liquid and can exist as a supercooled liquid at temperatures far below the melting temperature. The lowest measured temperature for supercooled liquid water is about -92 ◦C at 2 kbars.12
1.1. Thermodynamic anomalies As already mentioned, water displays a temperature of maximum den- sity (TMD) in the liquid phase at approximately 4◦C at the atmospheric pressure. As a consequence, the isobaric thermal expansion coefficient α (1/V)(∂V/∂T) , vanishes atT 4 ◦C and is negative below 4◦C P ≡ P ∼ where the volumeV expands upon cooling at constant pressureP . The ab- solute value α increases as a power law in the supercooled region.13 Ther- | P | modynamic relations show thatα P is proportional to the cross-fluctuations ofV and the entropyS. Hence, below the TM D at anyP , water volume and entropy are anti-correlated. This is just the opposite of what happens for the majority of the liquids. A consequence of this anomaly is, for the Clausius-Claipeiron relation, the negative slope of the melting line of water, Pm(T). Other anomalous thermodynamic functions of water are the isothermal compressibilityK (1/V)(∂V/∂P) 14 and the isobaric specific heat T ≡ − T C (∂H/∂T) ,15 whereH is the enthalpy of th e system, related to P ≡ P fluctuations in volume and entropy, respectively. For a normal liquidK T andC P tend to 0 upon cooling, while in the case of water they reach a ◦ minimum value at a certain temperature (T 46 C forK T andT 35 ◦ ∼ ∼ C forC P , at atmospheric pressure) and then increase for decreasingT. Hence, both fluctuations ofV andS increase upon cooling , at variance with normal liquids. We remind here that a TMD at constant pressure is revealed also in other substances: by experiments in Ga,16 Bi,17 Te,18 S,19,20 Be, Mg, Ca, 21 11 22–24 Sr, Ba, SiO2, P, Se, Ce, Cs, Rb, Co, Ge and by simulations on SiO2, 6 22 S, BeF2. As for the case of water, also for these substances the TMD implies anomalies in thermodynamic response functions. September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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Fig. 1. Scheme of water confined between hydrophobic plates of sizeL. The distance between plates inh≈0.5 nm.
wM K (T) M T CP (P) wM M αP (P) KT (T) 1 TMD M |α|P (P) M ξ (P) A 0,5 TminD wM |α|P (P) wM
Pressure P m K (T) K (T) 0 T T wM LIQUID CP (P) GAS -0,5 0 0,2 0,4 0,6 Temperature T
Fig. 2. Phase diagram for a mono-layer of liquid water (N = 104 molecules) nano- confined between hydrophobic walls separated by a distanceh≈0.5 nm. The loci, described in the text, are marked by full lines for the maxima, dashed lines for minima, and symbols, with labels nearby or in the legend. The liquid-to-gas spinodal (the lowest continuous line) delimits the region of stability of the liquid with respect to the gas phase. The LLCP (large circle with label “A”) is at the end of the LLPT (thick black) line. The indices “m”, “wM” and “M” refer, respectively, to minima, weak maxima and strong maxima of the corresponding thermodynamic quantities. Error bars are of the order of the swaying of the lines or the size of the symbols. September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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1.2. Dynamic anomalies A characteristic dynamic anomaly is the non-monotonic behavior of the diffusion constantD as function ofP along isotherms. In a normal liquid D decreases when the density or the pressure are increased. Anomalous liquids, instead, are characterized by a region of the phase diagram whereD increases upon increas ing the pressure at constant tem- perature.25 In the case of water, experiments show that the normal behavior ofD is restored only at pre ssures higher thanP 1.1 kbar at 283 K. 26 ∼
1.3. Structural anomalies, polymorphism and polyamorphism A typical structural anomaly is the non-monotonic behavior of structural order parameters of the system as a function ofP . Normal liquids tend to become more structured when compressed. The molecules adopt pref- erential separation and a certain orientational order. This ordering can be described by two order parameters: a translational order parameter and an orientational order parameter. The higher is the value of these param- eters, the higher is the order of the system. Therefore, for normal liquids these parameters increase with increasing pressure or density at constant temperature. Anomalous liquids, instead, show a region where the system becomes more disordered as the pressure increases, leading to lower values of the structural order parameters. This is emphasized in molecular dynamics sim- ulations for water27 and for silica.24 Another anomalous structural property is the occurrence of more than one crystal phase, a property called “polymorphism”. Water, carbon (with graphite, graphene and diamond), phosphorous (with black and red phos- phorous) and hydrogen28 are examples of polymorphic substances. In par- ticular, water has at least sixteen crystalline phases some of them stable only at high pressure. However, water has also with several amorphous (glassy) states,29–34 hence is called also “polyamorphic”. A glass is a frozen liquid state, metastable with respect to the crystal but possibly stable for time scales as long as the life-time of our civilizations. Indeed, we can still appreciate man-made glasses of ancient times, or amber stones preserving prehistorical insects. Experimental evidences of polyamorphism have been reported in many substances,35 including hydrogen,36 phosphorous,8–10 triphenyl phos- September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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phite,37–39 yttrium oxidealuminum oxide melts40 and stannic iodide.41 In the case of water, it has been found in experiments at low temperature and low pressure a low-density amorphous (LDA) ice.42 Upon increasing the pressure LDA transforms to high-density amorphous (HDA) ice,29 and upon further increasing of pressure a similar transformation has been observed from HDA to very-high-density amorphous (VHDA).43–45 As we discuss later, the presence of several amorphous states could be related to the occurrence of a metastable liquid-liquid phase transition.
2. Origin of water anomalies Different scenarios have been proposed to explain the origin of water anoma- lies. The stability limit scenario46 hypothesizes that the limit of stability of superheated liquid water merges with the limit of stretched and super- cooled water, giving rise to a single locus in theP T plane, with positive − slope at highT and negative slope at l owT , in the supercooled reg ime. The reentrant behavior of this locus would be consistent with the anomalies of water observed at higherT . As discussed by Debe nedetti, thermodynamic inconsistency challenges this scenario.47 The liquid-liquid critical point (LLCP) scenario48 supposes a first order phase transition in the supercooled region between two metastable liquids at different densities: the low-density liquid (LDL) at lowP andT , and the high-density liquid (HDL) at highP andT . The phase transition line has a negative slope in theP T plane and ends in a c ritical point. Nu- − merical simulations for several models for water are consistent with this scenario.48–61 The scenario is consistent also with several experiments for water, see for example,33,62–64 and for other systems, including silicon,65 phosphorous,8–10 triphenyl phosphite,37–39 yttrium oxidealuminum oxide melts40 and stannic iodide.41 Nevertheless, objections to this scenario have been raised,66 although they seems to apply only to a specific category of models.56–61,67 The singularity-free scenario68 focuses on the anti-correlation between entropy and volume as cause of the large increase of response functions at lowT and hypothesizes no hy drogen bond (HB) cooperativity. The scenario predicts lines of maxima in theP T for the response functi ons, similar to − those observed in the LLCP scenario, but shows no singularity forT> 0. Recent calculations show that the singularity-free scenario is a limiting case of the LLCP scenario with the LLCP atT = 0. 69 The critical-point-free scenario70 hypothesizes an order-disorder transi- tion, with a possible weak discontinuity of density, that extends toP<0 September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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and reaches the supercooled limit of stability of liquid water. This scenario would effectively predicts no critical point and a behavior for the limit of stability of liquid water as in the stability limit scenario. Stokely et al. have shown recently that all these scenarios have a com- mon physical mechanism and that two key physical quantities determine which of the four scenarios describes water: (i) the strength of the direc- tional component of the hydrogen bond and (ii) the strength of the co- operative (many-body) component of the hydrogen bond.69 Furthermore, estimates from experimental data for these two parameters of the hydrogen bond lead to the conclusion that water should have a LLCP at positive pressure.69 Nevertheless, to date experiments for bulk liquid water able to discrimi- nate between various models are not conclusive yet: the experimental reso- lution time is still bigger with respect to the homogeneous nucleation time and the system inevitably crystallize. As we will see in the next section, the confinement in nano-structures or nano-pores is, in some cases, a way to prevent the crystal formation.
3. A many-body model for confined water A variety of statistical models have been proposed in order to reproduce the main features of liquid water, including the anomalies. Water molecule are often treated as rigid and the water-water interaction is modeled combining Coulomb interactions of a finite number of point charges and Lennard-Jones potential between the centers of mass of different water molecules. These pair potentials reproduce water anomalies with fair agreement, but do not succeed in reproducing all the properties. For example, many of them fail in reproducing the crystal phases of water.71 However, the real problem with these models is that they are computationally expensive due to the long range Coulomb interaction. This problem is particular relevant in simulations of biological processes where macromolecules are surrounded by millions of water molecules. To overcome this problem a possible way is to consider coarse-grained models for water.52,72–76 In particular, these models can be used to study nano- confined water in extreme conditions and to compare with experiments. The study of nano-confined water is of great interest for applications in nanotechnology and nano-science.77 The confinement of water in quasi-one or two dimensions (2D) is leading to the discovery of new and controversial phenomena in experiments77–80 and simulations.78,81–84 Nano-confinement, both in hydrophilic and hydrophobic materials, can keep water in the liquid September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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phase at temperatures as low as 130 K at ambient pressure.79,80 At these temperaturesT and pressuresP experiments cannot p robe liquid water in the bulk, because water freezes faster then the minimum observation time of usual techniques, resulting in an experimental “no man’s land”.33 Nevertheless, new kind of experiments and numerical simulations can access this region, revealing interesting phenomena in the metastable state.48 Here, we will describe in detail the model proposed by Franzese and Stanley in 2002.52,73–76,85,86 The simplest approximation of nano-confined water is a monolayer of water confined between two plates87 (see Fig. 1). We coarse grain the spa- tial distribution of the molecules dividing the system into cells, each one occupied at most by one water molecule, and introduce a density field, ne- glecting the position of the water molecules inside the cell. For each cell i we associate an occupation numbern i = 1 if the local (dimensionless) densityv 0/v >0.5 (liquid-like configura tion), otherwisen i = 0 (gas-like configuration). Here,v is the cell volume and t he minimum volume of a cell corresponds to the hard core volumev 0 of a water molecule. Isotropic and long-range interaction (van der Waals attraction and hard core repulsion) in the system are represented by the Lennard-Jones potential.
r 12 r 6 H � 0 0 , (1) VW ≡ − r − r ij � ij ij � � � � � �
wherer ij is the distance between moleculesi andj and the sum is per- formed over all the neighboring molecules up to a cutoff distance at about twenty shells. This term depends only on the relative distance between two molecules and represents the isotropic part of the interaction. In order to describe the HB interaction, which is directional, we in- 68 troduce the variablesσ ij = 1...q for each occupied sitei facing the cell j. Assuming that a water molecule can form up to four HBs, we fix to four the number of variablesσ ij for each cell. Variablesσ ij are introduced to account for the number of bonding configurations accessible to a water molecule. The state of a water molecule is completely determined by the values assumed by the four variablesσ ij. The condition for two first neigh- bor molecules to form a HB isσ ij =σ ji. To take into account correctly the entropy loss due to the formation of a new HB, we consider that HB is broken if the angle HOO� deviates more than 30◦ from the linear bond. Therefore, we considerq = 180 ◦/30◦ = 6 states, where only 1/6 corresponds to the formation of a HB. This covalent HB interaction is represented by September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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the Hamiltonian term
H J n n δ , (2) HB ≡ − i j σij ,σji
H J δ , (3) Coop ≡ − σ σikσil i � (�k,l)i
whereJ σ > 0 is the characteristic energy of this cooperative component. The formation of a HB leads to an open structure that induces a local increase of volume per molecule. This effect is incorporated in the model by considering that the total volume of the system depends linearly on the number of HBs. So the volume change is
V V +N v , (4) ≡ 0 HB HB wherev is the increment due to the HB, andV Nv forN water HB 0 ≡ 0 molecules. The total enthalpy of the system is
H U+H +H +PV=U (J Pv )N JN +PV , (5) ≡ HB Coop − − HB HB − σ 0 where the total number of HB is
N n n δ ,( 6) HB ≡ i j σij ,σji
N δ , (7) σ ≡ σikσil i � (�k,l)i is the total number of HBs optimizing the cooperative interac- tion.52,69,73–76,85,87,89 In the following we adopt the parameters J/�=0.5, Jσ/�=0.05 andv HB/v0 = 0.5, consistent with the values that can be deduced from experiments.69 September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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4. Simulation details We simulate the model at constantT andP using an efficient clus ter al- gorithm89,90 that allows us to equilibrate the system very fast in the su- percooled region of the phase diagram, also for those combinations ofT andP where water exhibits glassy dynamics. To equilibrate our simula- tions, we implement an annealing protocol starting at high temperature and slowly decreasingT at constantP . The thermodynamic equilibrium is verified by checking that the calculation of the isothermal compress- ibilityK (T) (∂ ln V /∂P) , where V is the average volume from T ≡− � � T � � Eq. (4), and the isobaric specific heatC (P) (∂ H /∂T) , where H is P ≡ � � P � � the average enthalpy from Eq. (5), converge to their calculations from the fluctuation-dissipation relation. A Monte Carlo step is by definition the sequence of 4N + 1 trials of updating the 4N variablesσ and the volume of the system, in a random sequence. The trial change modifies the volume of a small random amount δV . We adjustδV to keep a constant acc eptance ratio 40%. 91,92 � In order to characterize the phase diagram of liquid water we simulate 104 different thermodynamic state-points for a wide range ofT andP ∼ with statistics of 5 10 6 independent calculations for systems withN= × 2500,..., 40000 water molecules,25,76,89,90,93–96 In real units the simulated range includes temperatures as low asT= 120 K and pressures as high asP=0.4 GPa. 93 Here, we adopt the following units: 4�/kB forT and 4�/v 0 forP , withk B the Boltzmann constant.
5. Results about the polyamorphism and the thermodynamics 5.1. Phase diagram The phase diagram is reported in Fig. 2. We find the liquid-to-gas spinodal atP< 0 and lowT , identifying the stabili ty limit of the liquid phase with respect to the gas phase. Within the liquid region we find along isobars the temperatures of maximum density (TMD). At highP the TMD line is reentrant, while at lowP it approaches the spin odal, without crossing it. At its lowest pressure the TMD line merges with the line of tempera- tures of minimum density (TminD) along isobars. This is consistent with experiments97 and other models.98,99 We calculate the isothermal compressibilityK T (T ) the isobaric expan- sivityα (P) (∂ ln V /∂T) along isotherms, and the isobaric specific P ≡ � � P heatC P (P ) along isotherms. For each quantity we find two maxima along September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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isobars at lowP . A broader and weak er maximum at higherT, and a sharper and stronger maximum at lowerT. 95 We find that the loci of weak maximaα wM and minima α wM of ther- P | | P mal expansivity along isotherms are both reentrant in pressure at lowT . At highT they converge toward each other forming a single locus of extrema that changes from maxima to minima where dP/dT= along the locus. ∞ This locus coincides within the error bars with the extrema of compress- wM m ibility along isobars, weak maximaK T and minimaK T , with the locus of extrema ofK T changing from maxima to minima where dP/dT = 0 along the locus. All these results are consistent with the thermodynamic relations.68,98,99 wM This consistency holds also for α P intersecting the TMD line in its 68 | | wM turning point at highT, and forC P , locus of weakC P maxima along isotherms, bending toward the point where the TMD line and the TminD line merge.99 M Finally, we find that the locus of strongC P maxima (CP ) coincides with the loci of stronger maxima α M and stronger maximaK M. This | P | T last relation among the loci of strong maxima has been found only within the framework of this model,86,100 likely because for other models is very expensive the computational cost to explore the thermodynamics of the metastable states at such a lowT , where the dynamics i s glassy,57 at low P , while at highP the strong and week m axima merge into each other. All the loci of the maxima of the response functions converge towards the point with the label “A” in Fig. 2. Moreover, all the maxima along these loci increase in their values by approaching the point “A”.
5.2. Polyamorphism and the liquid- liquid critical point The point “A” is at the end of a line along which we observe (i) a sudden change in the density of the liquid and (ii) a decrease of the maxima of the response functions for increasingP . These observations ar e consistent with the identification of this line with a first-order phase transition line separating two liquids with different densities, the HDL at highP andT and the LDL at lowP andT , consistent with the p resence of polyamorphism in water. The first-order liquid-liquid phase transition (LLPT) line ends in the state-point “A” where all the response functions diverge, that we, therefore, interpret as a liquid-liquid critical point, as in the LLCP scenario described in section 2. Confirmations of this interpretation come from the study of the correlation length and the critical exponent, discussed in the next sections. September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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5.3. The correlation length and the Widom line The increase of the response functions at the LLCP in “A” of Fig. 2 is related to the increase of fluctuations and this, in turn, is related to the increase of the correlation lengthξ. It is, therefore, interes ting to calculate the spatial correlation functionG(r ) σ (�r )σ (�r ) σ 2 to estimate il ≡� ij i lk l �−� ij � ξ, defined as the characteristic length over whichG(r) decays to zero. We find that forP below the LLCP in “ A”,G(r) decays as an ex- ponential with a characteristic correlation lengthξ. ForP approaching the LLCP,G(r) is better approximated by a power-law decay with an exponen- tial prefactor from whichξ can be extracted. At th e LLCP, the exponential prefactor approaches a constant leaving the power-law as the dominant contribution for the decay, corresponding toξ becoming of the order of the system size. All these observations confirm that the state point in “A” is, indeed, a critical point. Furthermore, the calculation ofξ near the LLCP allows us to gain an extremely relevant insight. We observe that forP
5.4. The critical region For a liquid-gas critical point it has been shown that an order parameter (o.p.) defined based solely on the difference in density between the two coexisting phases is not able to give rise to an o.p. distribution that is symmetric with respect to the ordered and disordered phases, as required at the critical point. The mixed-field finite-size scaling theory102–105 solves the problem by adopting as o.p. M ρ+ su, (8) ≡ September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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Fig. 3. The size-dependent probability distributionP N for the rescaled order parameter x, calculated for size-dependent critical temperatureT c(N) and pressureP c(N), has a symmetric shape that approaches continuously (fromN = 2500, symbols at the top atx = 0, toN = 40000, symbols at the bottom) the limiting form for the 2D Ising universality class (full line), maximizing the difference with the 3D Ising universality class (dashed line). Lines connecting the symbols are guides for the eye. Error bars are smaller than the symbols size.
where the densityρ represents the leading term,u is the energy density ands is the field mixing par ameter. Here, bothρ andu are expressed in internal units in such a way to be dimensionless. Within this theory,102–105 the probability distributionP (M) ˜p (x) N ∝ d ofM at the critical point sc ales as an universal function ˜p ofx B(M d ≡ − Mc). The function ˜dp is characteristic of the universality class of the Ising model ind dimensions and is the same for all the models belonging to the same universality class. The quantitiesB andM c are adjusted in such a way thatP N (M) has zero mean and unit variance. In particular, it can be shown thatB a −1N β/dν , whereβ> 0 is the critical exponen t ofM (T T) β, ≡ M ∼ c − ν > 0 is the critical exponent ofξ T T −ν ,T the critical temperature, ∼| − c| c anda M is a non-universal system-dependent parameter. Both exponentsβ September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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andν are defined by the uni versality class. As observed by other authors,106 the hypothesis of the mixed-field finite- size scaling theory hold in general for any fluid-fluid phase transition at a critical point. We, therefore, apply these theory also to the LLCP and the LLPT and adopt as o.p. the Eq. (8), wheres is found as explained i n the following.
Fig. 4. The size-dependent critical temperatureT c(N) (a) and pressureP c(N) (b) at the LLCP, agree with the power low functionN −(θ+1)/dν using critical exponents of Ising universality class ind = 2 dimensions:θ = 0,ν =1andβ=1/8. From these data we extract, by fitting in the thermodynamic limitN→∞,T c �0.0597 andP c �0.554. Continuous lines are best fits results. (c) We compare here the normalization factorB(N) (symbols) with the power law function∝N β/dν predicted by the theory (dashed line), finding a good agreement for large sizes. No fit is performed in this case and a systematic deviation from the predicted behavior is observed for small sizes.
For each system sizeN [2.5 10 3, 40 10 3], we simulate 300 state ∈ × × ∼ points, each for 100 independent runs o f 106 MC calculations, in a range ∼ ofT andP with 0.033 T 0.065 and 0.01 P 0.90, and use the ≤ ≤ ≤ ≤ multiple histogram reweighting method107 to extrapolate the calculations for intermediate state points. By tunings for these state points, we find that, for eachN, the distributionP N (x) has a symmetric shape with respect September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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tox = 0 whens=0.25 0.03 andT andP are in the vicinity of t he state ± point “A” (Fig. 3). We find that the resulting size-dependent critical parametersT c(N), Pc(N) and the size-dependent normalization factorB(N) follow in fair agreement the expected finite-size behaviors with the 2D Ising critical ex- ponents102–105 (Fig. 4). From the finite-size analysis we extract the asymp- totic valuesT = 0.0597 0.0001 andP = 0.554 0.003, consistent with c ± c ± the state point “A”. However, the behavior ofB(N) (Fig. 4c) clearly disp lays a systematic deviation from the predicted values for small sizes. A systematic analysis of the finite size affects of our results has been performed in,86 finding the surprising results that by decreasing the wall size with respect to the monolayer thicknessh, the LLCP is better de scribed by the 3D Ising model universality class. We ascribe this result to the strong cooperativity and the low coordination number of the hydrogen bond network.86
6. Study of the structural properties The many-body model described in section 3 successfully describes the thermodynamics of water under confinement and predicts the polyamor- phism at lowT . However, it coarse-gra ins the coordinates of the molecules, precluding the possibility to reproduce the details of the structure of the system. In particular, it does not allows to define the crystal phase.
6.1. The many-body model for fluid and crystal phases In this section we introduce a modification of the model that includes ex- plicitly the particle coordinates. As a consequence, the new formulation al- lows the calculation of the radial distribution function and the occurrence of the crystal-fluid phase transition, in addition to the properties already described by the previous sections. To this goal we add to the enthalpy in Eq.(5) a three-body repulsive interaction between a central moleculei and its H-bonded neare st neighbors k, l,
H J δ δ Δ(θi ), (9) Ang ≡ θ σikσki σilσli k,l i � (�k,l)i that is different from zero only if both the possible H-bonds are present. i Here,J θ > 0 is a characteristic interaction energy,θ k,l is the angle formed September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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by the centers of the three molecules and 1 Δ(θ) [1 + cos(4θ π)] (1 0) ≡ 2 − is a positive-defined function with minima (Δ = 0) at angles 90◦ and 180◦. For these two specific values of theθ it has been shown, by molecular dy- namics simulations of an atomistic model,81,82 that a water monolayer can form crystalline phases at lowT . In the following we ad opt the parameters J/�=0.75,J σ/�=0.1,v HB/v0 = 0.5 andJ θ/�=0.1. Here, the values for J/� andJ σ/� are larger than those adopted in the previous sections. The modification of these parameters does not change qualitatively the phase diagram of the fluid phases, but allows us the better calculate the properties of the crystal phases.
6.2. Simulation details We perform Monte Carlo simulations using the Metropolis algorithm. Here a Monte Carlo step is by definition the sequence of 5N +1 trials of updating theN molecule coordinates �r i, each free to move within a cell, the 4N variablesσ and the volume of the system, all choose in a random sequence. Each cell can be at most occupied by one molecule. To equilibrate the system we check the validity of the fluctuation- dissipation relations, as described for the model in the previous sections. Since we use periodic boundary conditions, in order to allow the equilibra- tion of an arbitrary crystal, we allow the change of the volume aspect ratio. We choose randomly theX orY direction and attempt a trial change of the box lateral sizesL X orL Y of a small random amount δL. We adjust δL to keep a constant acceptance ratio 40%. 91,92 �
7. Results about the polymorphism 7.1. Radial Distribution Function In order to study structural properties of our system, we calculate the radial distribution function in two dimensions,
1 1 N g(r) δ(r r ), (11) ≡ 2πr Nρ − ij i=0 � � �i=j wherer �r �r is the distance between two moleculesi andj, andr ij ≡ | i − j | is the distance in units ofr v /h, whereh 0.5 nm is the distance 0 ≡ 0 � between the confining hydrophobic walls. � September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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Fig. 5. Radial distribution functiong(r) of the 2D projection of the coordinates of the confined water monolayer at different temperatures forP=0.1 (upper panel), and for P=1.5 (lower panel). In bot h panels theg(r) changes from a low-s tructured function of a fluid phases at highT=0.30 to a highly-structu red function of a crystal phase at low T=0.09. For the same value s ofT , theg(r) at the two values ofP are quite different. In particular, theg(r) of the crystal phase s atT=0.09 have peaks in diffe rent positions, emphasizing that the two crystals are different and that the system is polymorphous as discussed in the text. September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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We find thatg(r) calculated at differentT andP (Fig. 5) emphasizes the occurrence of fluids and crystals with different symmetries. This observation is consistent with the occurrence of polyamorphism and polymorphism in the confined water monolayer. Next we characterize the symmetries of the different phases.
7.2. Structural Order Parameters
Fig. 6. Configurations for the coarse-grained model defined in section 6.1 at different temperatures forP=0.1 (panels a, b, c), and forP=1.5 (panels d, e, f). The orienta- tional and translational order are quantified using the corresponding order parameters |Φ4| and|Ψ G|. Two different solid phases at low temperature are found having square and triangular symmetry at low and high pressure, respectively. HereT andP are as in Fig. 5.
We adopt a local bond-orientational order parameterφ 4, similar to that introduced by Nelson and Halperin,108 which is a function of the state of September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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the system, n 1 i φ ei4θij , (12) 4,i ≡ n i j=1 � and its global average
1 N Φ φ .( 13) 4 ≡ N 4,i � i=1 � � The angleθ ij is the angle formed by the vector going from the molecule i to the moleculej with respect to an ar bitrary reference axis andn i is the number of nearest neighbors of the moleculesi. The quantity Φ 4 gives information about the amount of orientational local order in the system, being Φ 0 for random (fluid) configurations, and Φ 1 for crystalline | 4| � | 4| � configurations with 90◦ symmetry. Next, we define the translational order parameter considering � ψ e iG·�ri , (14) G,i ≡ and its average
1 N Ψ ψ ,( 15) G ≡ N G,i � i=1 � � where, taking as a reference a crystal with square symmetry, we choose � 2π √ G= a �eX , wherea=L X / N is the lattice spacing and �eX the unit vector along the coordinateX. The order parameter ΨG quantifies the average translational order in the system and is Ψ 1 only when the | | G � system is highly structured.109 We find110 that at low pressure and lowT the systems crystallize s with long range translational and orientational order with square symmetry due to the formation of a fully connected hydrogen-bond network (Fig.6,a). At the same pressure and higherT the systems looses its long-range trans- lational order and is characterized by a short-range orientational order (Fig.6,b) as in a hexatic phase.111,112 At higherT the system is in a diso r- dered (fluid) state (Fig.6,c). By increasing the pressure, the system crystallizes at lowT with a tri- angular symmetry due to the breaking of the hydrogen bonds (Fig.6,d). In this case the crystal is better characterized by order parameters with triangular symmetry Φ6 and ΨG with definitions similar to those given in Eqs. (13, 15). By increasingT , also a this pressure t he system looses its long-range order, keeping the short-range order (Fig.6,e) as in another September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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hexatic phase.111,112 At even higherT the system melts into a disordered (fluid) state (Fig.6,f) with a density that is larger than the density of the fluid at lower pressure. All these structural transformations are well reflected by the behavior of g(r) seen in Fig. 5. We, therefore, conclude that a confined water monolayer displays polymorphism at lowT , and polyamorphism at higherT , as a function of the pressure. A more quantitative analysis, with the definition of the limit of stability of the different phases, is performed in Ref. 110.
8. Conclusions Our results show that a water monolayer confined between hydrophobic parallel walls separated byh 0.5 nm has qualitatively the same phase ≈ diagram of bulk water at highT . At lowT , where bulk water fre ezes, confined water can still be enough fluid to allow the coexistence of two liquids phases, HDL and LDL, with different density, energy, entropy and structure. Under these conditions, confined water is a polyamorphous. The two liquids are separated by a LLPT ending in a LLCP. This crit- ical point belongs to the 2D Ising universality class for a monolayer in the thermodynamic limit, i.e. when the walls lateral sizeL h. Our analysis, � as shown in Ref. 86, show the intriguing result that for L/h 50 the sys- ≤ tem is better described by the 3D Ising universality class. Surprisingly this suggests that water, under strong confinement, could be better described by bulk-like critical behavior. We ascribe this property to the high cooper- ativity and low coordination number of the hydrogen bond network.86 By introducing the molecules coordinates within the coarse-graining cells of the model,110 we are able to describe also the crystal phases of confined water. We find that the system is polymorphous, reminiscent of the large number of crystal phases of bulk water. Our analysis110 allows us to understand the possible origin of different interpretations of numerical results coming from different models.66,67 In particular, in Ref. 110 we show that the LLPT is hindered by inevitable crystallization when the three-body interaction among the water molecules is stronger than the interaction coming from the remaining many-body con- tributions to the water-water coupling. Therefore, a model where the three- body interaction is dominant, as in the mW-water case,67 would favor the crystallization over the liquid-liquid coexistence, while in other models57 the contributions to the water-water coupling coming from four-body interac- tion and higher order interaction would favor the liquid-liquid coexistence over the crystallization. September 11, 2013 17:26 WSPC - Proceedings Trim Size: 9in x 6in perspectives
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Acknowledgments G. F. has the pleasure to thank H. E. Stanley and L. Lucena for sharing with him and the community worldwide their perspectives in Statistical Physics and Complex Systems for the next decade. We also thank M.C. Barbosa, M. Bernabei, S.V. Buldyrev, F. Bruni, S.-H. Chen, P. Debenedetti, P. Kumar, F. Leoni, J. Luo, G. Malescio, F. Mallamace, M. I. Marqus, M. G. Mazza, A.B. de Oliveira, S. Pagnotta, D. Reguera, F. de los Santos, F. Sciortino, H. E. Stanley, F. W. Starr, K. Stokely, and E. G. Strekalova, and P. Vilaseca for helpful discussions. We thank for support the Spanish Ministerio de Ciencia e Innovaci´onGrant FIS2009-10210, the Spanish Ministerio de Economia y Competitividad Grant FIS2012-31025, the Generalitat de Catalunya Grant 2010 FI-DGR, and the European Commission Grant FP7-NMP-2010-EU- USA.
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