Dielectric Constant and Proton Order and Disorder in Ice Ih: Monte Carlo Computer Simulations

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Chemistry Faculty Publications Department of Chemistry

5-2003

Dielectric constant and proton order and disorder in ice Ih: Monte Carlo computer simulations

Steven W. Rick University of New Orleans, [email protected]

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Recommended Citation Steven W. Rick and A. D. J. Haymet. 2003. " Dielectric constant and proton order and disorder in ice Ih: Monte Carlo computer simulations." Journal of Chemical Physics 118 (20): 9291-9296.

This Article is brought to you for free and open access by the Department of Chemistry at ScholarWorks@UNO. It has been accepted for inclusion in Chemistry Faculty Publications by an authorized administrator of ScholarWorks@UNO. For more information, please contact [email protected]. JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 20 22 MAY 2003

Dielectric constant and proton order and disorder in ice Ih: Monte Carlo computer simulations Steven W. Ricka) Department of Chemistry, University of New Orleans, New Orleans, Louisiana 70148 and Chemistry Department, Southern University of New Orleans, New Orleans, Louisiana 70126 A. D. J. Haymetb) Department of Chemistry, University of Houston, Houston, Texas 77204-5003 ͑Received 10 December 2002; accepted 26 February 2003͒ The dielectric properties of ice Ih are studied using a Monte Carlo algorithm for sampling over proton conﬁgurations. The algorithm makes possible the calculation of the dielectric constant and other properties of the proton-disordered crystal. Results are presented for three classical models of water, two commonly used nonpolarizable models ͑SPC/E and TIP4P͒ and a polarizable model ͑TIP4P-FQ͒.©2003 American Institute of Physics. ͓DOI: 10.1063/1.1568337͔

I. INTRODUCTION sition occurs at 71.6 K, apparently independent of the type and concentration of the dopant.4 The high dielectric constant of ice Ih represents one of A general method for generating proton positions in the the many anomalous properties of the condensed phases of ice lattice is first to identify a set of water molecules which water.1,2 At the coexistence temperature at 1 atm pressure, form a hydrogen-bonded loop and then to shift collectively ice has a larger dielectric constant than liquid water, even the protons along the loop. This approach was first used by though, as is well known, it has a lower density. The high Rahman and Stillinger, who treated all proton configurations dielectric constant is indicative of the disorder of the protons satisfying the ice rules as energetically equivalent.5 Subse- present in ice Ih ͑as well as some other ice phases͒. The quent studies of two-dimensional ice by Yanagawa and mechanism for proton rearrangement in ice involves both Nagle6 and ice Ih by Barkema and de Boer7 used the Monte orientational defects ͑the Bjerrum D and L defects͒ and ionic Carlo method to sample proton configurations according to defects.3 In pure ice, these defects are present in extremely their energies. These studies placed the oxygen and hydrogen low concentrations, one in 5ϫ106 water molecules for Bjer- atoms on lattice sites, with no lattice vibrations or distor- rum defects and 105 times less for ionic defects. Computer tions. The only degrees of freedom not fixed in these studies simulations typically use system sizes of 100–1000 mol- were the protons. Here we present an algorithm for generat- ecules and therefore cannot include these defects at the nor- ing proton positions, which is general enough to be com- mally occurring concentrations. In this work, we present an bined with sampling over the other degrees of freedom. In algorithm for sampling efficiently over proton configura- Sec. II we describe the method and give the details of the tions, which yields the calculation of the dielectric constant simulations. In Sec. III, we present the results for the dielec- of ice. tric constant and proton order parameters of ice for three The water molecules are arranged according to the ice water models, and in Sec. IV, we summarize our results. rules: water molecules are present as neutral H2O and each molecule makes four hydrogen bonds with its four nearest neighbors, two as a hydrogen bond donor and two as an II. METHODS: THE MOVE ALGORITHM acceptor. If all proton configurations satisfying the ice rules To sample over proton configurations with the proper are equally probable, then the resulting entropy difference Boltzmann weighting at a specified temperature, we have between the disordered and ordered states is in very good developed an algorithm involving two steps. The first step is agreement with the observed residual entropy of ice. How- to find a closed hydrogen bonded loop in the ice crystal using ever, all proton configurations are most likely not energeti- a random walk. The second step is to generate new proton cally equivalent. At temperatures below 100 K, it is believed positions, satisfying the ice rules, by changing the hydrogen that a transition occurs to a proton-ordered phase, represent- bond pattern. Our algorithm is described as follows: ing the lowest-energy proton configuration.4 The annealing ͑ ͒ times may be prohibitively long to observe the ordered phase 1 Find a closed hydrogen-bonded loop. ͑ ͒ of pure ice, but for ice Ih doped with alkali hydroxides, a Randomly select a molecule in the lattice i. ͑ ͒ which decrease the relaxation time, the order–disorder tran- b Randomly choose one of its four nearest neighbors j. ͑c͒ If j is a hydrogen bond donor to i, then randomly pick one of the two nearest neighbors of j which are a͒ Author to whom correspondence should be addressed. Electronic mail: hydrogen bond donors to j; otherwise, randomly pick [email protected] b͒Present address: CSIRO Marine Research, GPO Box 1538, Hobart one of the two nearest neighbors of j which are hy- TAS 7001 Australia. drogen bond acceptors to j.

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molecule is rotated. One method is to simply rotate the velocities of each atom along the position. In the standard Ver- let algorithm, the positions at the previous time step, rather than the velocities, are used together with the present positions and forces to generate the trajectory.10 Using the Verlet algorithm, the previous time step positions for all three at- ␪ oms are rotated by the same angle around the O– H1 axis ͑of the present position͒. This is equivalent to rotating the velocities and will not change the kinetic energy, or temperature, because only the direction, not the magnitude, is FIG. 1. Illustration of the Monte Carlo algorithm for generating new proton changed. However, the momentum will be changed and each

positions by rotation about the Oj –H1 j axis. accepted proton move will change the net momentum of the system. To keep the momentum of the system constant, a ͑ ͒ d Continue to walk randomly until the path connects small modification is required. The velocity of atom ␣ on back to any spot on the path, not necessarily the first molecule i is given by molecule. For finite-size systems, the walk will have ϭ͓ ͑ ͒Ϫ ͑ Ϫ⌬ ͔͒ ⌬ ͑ ͒ to cross itself at some point. For systems with peri- vi␣ ri␣ t ri␣ t t / t, 1 odic boundary conditions, the closed loop may be a ⌬ where t is the time step, ri␣(t) is the present position, at molecule displaced by the side length of the periodic time t, and r ␣(tϪ⌬t) is the previous position. If the net ͑ i box a periodic image of the original molecule in the momentum is to be conserved, then the contribution to the ͒ loop . momentum from the molecules which are being moved must ͑2͒ Generate new proton positions. be the same before and after the proton move. For a ͑a͒ For each molecule j on the N-molecule loop, rotate hydrogen-bonded loop with N molecules, the contribution to the molecule about the Oj –H1 j axis, where H1 j is the momentum from these N molecules is given by hydrogen atom which is not in the loop. The molecule N ␪ is rotated by an angle j . a ϭ a PN ͚ ͚ m␣vi␣ ͑ ͒ ␪ ϭ ␣ b The angle j depends on the position of OjϪ1 ,Ojϩ1 , i 1 ͑ ͒ Oj , and H1 j see Fig. 1 . It is equal to the angle N between the planes formed by OjϪ1 ,Oj , and H1 j and ϭ ͓ a ͑ ͒Ϫ a ͑ Ϫ⌬ ͔͒ ⌬ ͑ ͒ ͚ ͚ m␣ ri␣ t ri␣ t t / T, 2 ϭ ϭ ␣ Ojϩ1 ,Oj , and H1 j . For j 1, the OjϪ1 corresponds i 1 ϭ to ON , and for j N, Ojϩ1 corresponds O1 . ␣ where vi␣ is the velocity of atom of molecule i, ri␣(t) and Ϫ⌬ The move is accepted based on the standard Monte ri␣(t t) are the present and previous positions, respec- 8 Carlo or Metropolis algorithm, with an acceptance probabil- tively, m␣ is the mass of atom ␣, and ⌬t is the time step. The Ϫ (Ea Eb)/kT ity equal to Min(1,e ), where Ea is the energy be- superscript a indicates variables before rotation. After rota- fore and Eb is the energy after the proton move, k is the tion, the contribution to the momentum from the same N Boltzmann constant, and T is temperature. The many-proton molecules is moves can be combined with smaller amplitude moves of the N oxygen and hydrogen atoms using standard Monte Carlo or b ϭ b PN ͚ ͚ m␣vi␣ canonical ensemble ͑constant T,V,N) molecular dynamics. iϭ1 ␣ Either method is applicable, and we choose to use constant- N temperature molecular dynamics because it is more efficient ϭ ͓ b ͑ ͒Ϫ b ͑ Ϫ⌬ ͔͒ ⌬ ͑ ͒ 9 ͚ ͚ m␣ ri␣ t ri␣ t t / T. 3 for use with polarizable models. iϭ1 ␣ Step 1, for finding the hydrogen-bonded loop, is the b ϭ a same as methods presented previously for generating proton In order for PN PN , the previous positions for all atoms in configurations in ice.5–7 The previous studies all used oxygen the loop, after they have been rotated, must be shifted by ⌬ and hydrogen positions on an otherwise perfect lattice, with adding a constant R. The constant is given by no lattice vibrations or molecular rotations. In this case, gen- ⌬ ͑ b Ϫ a ͒ t PN PN erating the new proton positions simply involves moving the ⌬Rϭ , ͑4͒ M N proton along the line connecting the neighboring oxygen atoms. Due to thermal motion, however, the hydrogens will where M is the total mass of the water molecule. This modi- not necessarily be on the O–O line, as illustrated in Fig. 1, so fication will lead to conservation of the momentum, but will a different method for generating new proton positions is change slightly the kinetic energy of the system, at each ͑ ͒ ␪ required step 2 . The rotation angle j is about 120°, but accepted proton move. will vary due to lattice vibrations and nonideal values for the The proton move algorithm is both ergodic and satisfies lattice constant ratio c/a. Using a rotation angle equal to detailed balance, and therefore can sample over all proton 120° gives a vanishingly small acceptance ratio for the pro- configurations with the proper Boltzmann weighting. The er- ton moves and is not recommended. godicity of the loop algorithm has been demonstrated Combining the proton Monte Carlo moves with molecu- previously.7,11 In brief, any difference in hydrogen bonded lar dynamics requires a new set of velocities after each water configurations can be represented in terms of different

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hydrogen-bonded loops, and so all configurations can be reached in principle. An earlier study of proton disorder has demonstrated this same point, that all different proton configurations, which obey the ice rules, can be expressed as a number of different loops on the ice lattice.12 The detailed balance condition ensures that the probability of generating a configuration is proportional to its Boltzmann weighting. De- tailed balance equates the probability of being at a specific state a times the transition probability from a to a new state ⌸ b, ab , to the probability of being at a specific state b times the transition probability from b to a,or Ϫ Ϫ Ea /kT⌸ ϭ Eb /kT⌸ ͑ ͒ e ab e ba . 5 ⌸ For the many-proton move, ab can be written as the prod- ⌸ ϭ⌸L ⌸R ⌸L uct of three terms, ab ab abAab , where ab is the probability of generating the specific hydrogen-bonded loop FIG. 2. The probability of generating a loop of size N ͑diamonds͒ and the ͑ ͒ ⌸R ͑ ͒ step 1, above , ab is the probability for generating the new acceptance ratio of a move involving N molecules crosses as a function of proton positions in that loop ͑step 2͒, and A is the accep- N, shown on a logarithmic scale. The values are for the TIP4P-FQ model at ab ϫ ϫ tance probability for the move. The probability of generating 273 K with a 128-molecule 4 2 2 lattice. the loop in configuration a from the random walk is the same ⌸L as generating the loop in configuration b, and so ab ϭ⌸L ͑ ͒ ␪ ba see also Ref. 7 The rotation angle j is defined by which is an efficient method for keeping the charges close to the oxygen and hydrogen positions in such a way that the the ground-state minimum. Each time a many-proton move angle that would be used to generate the new proton posi- is attempted, the exact minimum-energy charges are found tions starting from state a is exactly minus the angle that for the configuration before and after the protons are ⌸R would generate the proton positions from b. Therefore ab changed. The simulations are performed over a range of tem- ϭ⌸R ba . The probability of generating a trial move is equal to perature, from 10 to 273 K. Each temperature is simulated ⌸ that of the reverse move. With this condition on ab , de- for 100 ns, except for some points at lower temperatures, tailed balance is satisfied if where more simulation data are required to obtain suffi- Ϫ Ϫ ciently small error bars. The TIP4P-FQ model was simulated Ea /kT ϭ Eb /kT ͑ ͒ e Aab e Aba , 6 for 200 ns at 50 and 100 K and TIP4P was simulated for 240 which the standard Metropolis acceptance criteria satisfies. ns at 50 K. In the simulations reported here, a many-proton move The acceptance ratio for the many-proton move depends was attempted every 100 molecular dynamics time steps. on temperature and on the water model. For TIP4P, the ac- The simulations were done using three water different water ceptance ratio ranges from 0.8% at 50 K to 5.2% at 240 K. models: the nonpolarizable SPC/E ͑Ref. 13͒ and TIP4P ͑Ref. For TIP4P-FQ, the acceptance ratio ranges from 0.2% at 100 14͒ models and the polarizable TIP4P-FQ ͑Ref. 9͒ model. K to 2.5% at 273 K. For SPC/E, the acceptance ratio at 200 We use a time step of 1 fs and the SHAKE algorithm for bond K is 1.3%. The acceptance ratio also depends on the size of constraints.10 The models used here are all rigid and classi- the loop. Figure 2 also shows the probability of generating a cal, but our proton move algorithm is sufficiently general to loop of size N. The smallest loop possible on the ice Ih be able to treat flexible or quantum mechanical15 models as lattice contains 6 molecules and the probability of generating well. Long-ranged interactions are treated using Ewald sums, an N-molecule loop through the random walk decreases as N with a screening parameter ␬ equal to 5/L, where L is the increases. ͑This probability will depend on the size of the smallest box side length, and 256 reciprocal space lattice system, depending on how many molecules it takes to reach vectors.10 The simulations used 128 molecules ͑except as a periodic image.͒ The acceptance ratio drops off approxi- noted͒ in the canonical (T,V,N) ensemble by coupling to a mately exponentially with N. Nose´–Hoover thermostat and used a orthorhombic unit cell. The lattice constants were chosen to yield a pressure of 1 atm. The original configuration for all simulations was taken III. RESULTS to be the proton-ordered, antiferroelectric structure of David- A. Dielectric constant son and Morokuma16 ͑except as noted below͒. This unit cell contains 8 molecules and is replicated 16 times ͑4inthex The static dielectric constant can be found from ͒ direction and 2 in the y and z directions to give the 128- 4␲ ͑ ⑀ϭ⑀ ϩͩ ͪ ͑͗ 2͘Ϫ͗ ͘2͒ ͑ ͒ molecule sample. The x, y, and z directions correspond to ϱ M M , 7 3VkT the a, b, and c lattice parameters.͒ For the TIP4P-FQ model, ͗ ͘ polarization is treated using fluctuating charges, namely, where M is the total dipole moment of the system and ¯ charges which change in response to different electrostatic indicates an ensemble average.10 For the polarizable model, 9 environments. The charges are updated in the molecular dy- the optical dielectric constant ⑀ϱ is set equal to 1.592, and namics simulation using the extended Lagrangian method, for the nonpolarizable TIP4P and SPC/E models, ⑀ϱ equals

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TABLE I. Dielectric constants and G factors as a function of temperature phase at these temperatures. This would indicate a proton- for the TIP4P, TIP4P-FQ, and SPC/E models of water. Numbers in paren- ordered phase transition, with a transition temperature be- theses represent 95% conﬁdence limits. tween 50 and 100 K for TIP4P-FQ and 10 and 50 K for

Model T ͑K͒ ⑀⑀Ќ ⑀ʈ G TIP4P. The transition temperature is in the range of the esti- 7,17,18 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ mates from other theoretical and experimental TIP4P-FQ 273 91 3 90 11 92 5 2.63 9 4,19–23 TIP4P-FQ 240 100͑5͒ 101͑7͒ 99͑13͒ 2.4͑1͒ studies, although the experimental data on the KOH- TIP4P 240 48͑1͒ 47͑4͒ 50͑3͒ 2.54͑5͒ doped ice indicate that the proton-ordered state is TIP4P-FQ 200 130͑16͒ 132͑29͒ 127͑13͒ 2.5͑3͒ ferroelectric.24,25 The apparent transition may just be due to TIP4P 200 57͑3͒ 58͑4͒ 57͑5͒ 2.5͑1͒ sampling problems at low temperatures. SPC/E 200 50͑4͒ 55͑1͒ 40͑4͒ 1.9͑2͒ TIP4P-FQ 150 165͑26͒ 173͑25͒ 149͑32͒ 2.2͑3͒ The polarizable TIP4P-FQ model has a higher dielectric TIP4P 150 78͑4͒ 77͑7͒ 80͑4͒ 2.6͑1͒ constant than the nonpolarizable models and is close to the TIP4P-FQ 100 213͑51͒ 192͑61͒ 256͑99͒ 1.7͑4͒ experimental values. The high-temperature values for ⑀ are TIP4P 100 105͑5͒ 104͑12͒ 108͑5͒ 2.3͑1͒ similar to the calculated dielectric constants for each of the ͑ ͒ ͑ ͒ ͑ ͒ TIP4P-FQ 50 1.71 3 1.73 3 1.68 1 0 models in the liquid phase.9,26–30 The better agreement with TIP4P 50 212͑25͒ 218͑45͒ 201͑27͒ 2.3͑3͒ TIP4P-FQ 10 1.70͑4͒ 1.74͑6͒ 1.63͑1͒ 0 the experimental ice dielectric may be due to the dipole mo- TIP4P 10 1.05͑1͒ 1.08͑1͒ 1.00͑1͒ 0 ment of the TIP4P-FQ model, which is about 3.0 D at 273 K,30 rather than 2.18 for TIP4P or 2.35 for SPC/E. Onsager and Dupuis31 and Rahman and Stillinger5 have both shown that a dipole moment of about 3.0 D is consistent with the 1. The average of M is zero and the ͗M͘2 term in Eq. ͑7͒ can experimental dielectric constant. This value is also consistent be omitted. The anisotropy of the dielectric constant is given with other theoretical predictions of the dipole moment of a by the components water molecule in ice.32–34 The statistical errors in the calculations are too large to 4␲ ⑀ ϭ⑀ ϩͩ ͪ ͑͗ ͘Ϫ͗ ͗͘ ͒͘ ͑ ͒ resolve differences in the parallel and perpendicular compo- ␣␤ ϱ M␣ M␤ M␣ M␤ . 8 3VkT • nents of ⑀. The measured values show that ⑀ʈ is greater than 3 The dielectric response is also commonly expressed in terms ⑀Ќ . From the calculated values, no clear trend about the of the correlation parameter G, which equals relative magnitudes of the two components is apparent and 2 ␮2 ␮ ͗M ͘/(Nmolec͗ ͘), where is the dipole moment of a wa- the anisotropy appears to be small. There is a disagreement ter molecule and Nmolec is the number of molecules in the in the reported experimental anisotropies, ranging from less system. than 1% to more than 20% at 0 °C.19 Theoretical estimates The calculated values of the dielectric constants and G find small anisotropies, less than 1% ͑Refs. 7, 35, and 36͒ are shown in Table I. In agreement with experiment, the ͑except for Ref. 5͒. The G factor is less than 3.0 and de- calculated dielectric constants decrease with temperature creases as the temperature is lowered. For a fully random ͑Fig. 3͒. At low temperatures, starting with 50 K for lattice, G is equal to 3.0.35,36 The perfect lattice simulations TIP4P-FQ and 10 K for TIP4P, the protons are frozen in the of Barkema and de Boer also showed that G decreases as the antiferroelectric ground state and no proton moves are ac- temperature decreases.7 The decrease in G indicates that the cepted. The low-temperature results suggest that for these protons are not distributed randomly. potential models the proton-ordered lattice may be the stable B. Order parameters Hydrogen bonds between two molecules can be charac- terized by a dihedral angle ␾ between the HOH angle bisec- tor of the hydrogen bond acceptor and the OH1 axis of the hydrogen bond donor (H1 is the hydrogen not involved in this hydrogen bond͒. For hydrogen bonds along the c axis, the dihedral angle can be 60° ͑these are termed oblique mirror37͒ or 180° ͑inverse mirror͒. For hydrogen bonds not along the c axis, the angle can be 0° ͑inverse center͒ or 120° ͑oblique center͒. The most stable configuration for the water dimer has ␾ equal to 180°. This configuration places the hydrogen not involved in the hydrogen bond away from the hydrogens of the other molecule. Based on the dimer energies, the inverse mirror and the oblique center configurations are lower in energy. However, as illustrated by Pitzer and Polissar, next-nearest-neighbor interactions in the ice lattice decrease the energy differences.17 We can define order parameters for the ice lattice based on the angle ␾ and define FIG. 3. The calculated dielectric constants of ice Ih for the TIP4P-FQ model X , X , X , and X as the fraction of hydrogen bonds ͑dashed line and diamonds͒, TIP4P model ͑dotted line and crosses͒, and im om ic oc SPC/E model ͑the square͒, compared to the experimental values from Ref. that are inverse mirror, oblique mirror, inverse center, and 17 ͑solid line͒. oblique center, respectively. Since there are four total hydro-

TABLE II. Hydrogen bond orientational order parameters, comparing the simulation results to the values for different lattices.

͑ ͒ ϩ Model T K Xim Xom Xic Xoc (Xom Xic)/4 Fully random lattice 0.333 0.667 1.0 2.0 0.417 Constrained random lattice 0.417 0.583 0.997 2.003 0.395 Antiferroelectric lattice 1.0 0.0 0.0 3.0 0 Ferroelectric lattice 1.0 0.0 3.0 0.0 0.75 TIP4P-FQ 273 0.379͑4͒ 0.621͑4͒ 0.942͑1͒ 2.058͑1͒ 0.391͑1͒ TIP4P-FQ 240 0.385͑4͒ 0.615͑4͒ 0.935͑7͒ 2.065͑7͒ 0.388͑1͒ TIP4P 240 0.368͑3͒ 0.632͑3͒ 0.950͑3͒ 2.050͑3͒ 0.396͑1͒ TIP4P-FQ 200 0.387͑4͒ 0.613͑4͒ 0.92͑1͒ 2.08͑1͒ 0.384͑4͒ TIP4P 200 0.374͑2͒ 0.626͑3͒ 0.952͑4͒ 2.048͑4͒ 0.395͑1͒ SPC/E 200 0.368͑8͒ 0.60͑2͒ 0.87͑2͒ 2.13͑2͒ 0.40͑2͒ TIP4P-FQ 150 0.409͑7͒ 0.591͑7͒ 0.91͑1͒ 2.09͑1͒ 0.374͑3͒ TIP4P 150 0.376͑4͒ 0.624͑4͒ 0.948͑5͒ 2.052͑5͒ 0.393͑2͒ TIP4P-FQ 100 0.45͑2͒ 0.55͑2͒ 0.87͑3͒ 2.13͑3͒ 0.35͑1͒ TIP4P 100 0.389͑3͒ 0.611͑3͒ 0.933͑9͒ 2.067͑9͒ 0.386͑2͒ TIP4P-FQ 50 1.0 0.0 0.0 3.0 0.0 TIP4P 50 0.42͑1͒ 0.58͑1͒ 0.91͑2͒ 2.09͑1͒ 0.373͑3͒ TIP4P-FQ 10 1.0 0.0 0.0 3.0 0.0 TIP4P 10 1.0 0.0 0.0 3.0 0.0

Ϫ gen bonds for each molecule and one is along the c axis, gen bonds. For TIP4P-FQ, ͗Eim͘ ͗Eom͘ equals 0.08 kcal/ ϩ ϩ Xim Xom must equal 1 and Xic Xoc must equal 3. The mol, and for TIP4P, it equals 0.03 kcal/mol. The values for Ϫ Ϫ fraction of high-energy hydrogen bonds can be deﬁned as ͗Eic͘ ͗Eoc͘ are 0.04 kcal/mol for TIP4P-FQ and ϩ Ϫ (Xom Xic)/4. 0.008 kcal/mol for TIP4P. The differences are smaller for The calculated order parameters as a function of tem- the TIP4P model. Both models are an order of magnitude perature are shown on Table II. Also shown are the values for smaller than the estimates of Pitzer and Polissar, which only various lattices: the fully random lattice, the random lattice take into account some of the nearest neighbor interactions.17 of Hayward and Reimers, generated with the ice rule, zero 38 dipole moment, and zero quadrupole moment constraints, C. Dependence on system size and initial conditions the antiferroelectric lattice of Davidson and Morokuma,16 and the ferroelectric lattice.24,25 Note that the Hayward– Previous calculations of the dielectric constant of liquid Reimers lattice has values that are slightly different from the water have been done for a variety of system sizes, including fully random lattice for the order parameters Xim and Xom . The constraints apparently cause Xom to not equal 2Xim . ͑The values are for the 6ϫ4ϫ4 lattice with 768 molecules.38͒ The values from the simulations show that the proton positions are not fully random and that the fraction of ϩ high-energy hydrogen bonds, (Xom Xic)/4, decreases with temperature. The order parameters approach the antiferroelectric and not the ferroelectric values. This trend is consistent with the results of Barkema and de Boer.7 We can write

Xim 1 Ϫ ͗ ͘Ϫ͗ ͘ ϭ e ( Eim Eom )/kT ͑9͒ Xom 2 and

Xic 1 Ϫ ͗ ͘Ϫ͗ ͘ ϭ e ( Eic Eoc )/kT, ͑10͒ Xoc 2

where ͗E␣͘ is the average energy of hydrogen bonds of type ␣, at a particular temperature and density, and the factor of 1/2 is from the fact that there are twice as many oblique as inverse hydrogen bonds. If the temperature dependence on the average energy is small, then a plot of ln(Xim /Xom) and ln(Xic /Xoc) versus 1/T should give a straight line, with the FIG. 4. The energy difference between the two types of hydrogen bonds: the slope indicating the energy difference between the two types ͓ ͔ ͑ ͒ ͓ ͔ ͑ ͒ ͑ ͒ ln Xim /Xom top and ln Xic /Xoc bottom as a function of inverse tempera- of hydrogen bonds see Fig. 4 . The slopes give a very small ture for the TIP4P-FQ ͑solid line and diamonds͒ and TIP4P ͑dashed lines energy difference between the two different types of hydro- and crosses͒ models of water.

Downloaded 29 Mar 2011 to 137.30.164.143. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 9296 J. Chem. Phys., Vol. 118, No. 20, 22 May 2003 S. W. Rick and A. D. J. Haymet

systems smaller than 128 molecules,39 and there is no ob- 1 N. H. Fletcher, The Physics of Rainclouds ͑Cambridge University Press, servable strong dependence on the system size. For the solid, Cambridge, England, 1962͒. 2 sampling over fluctuations of M requires accepting many N. Fletcher, The Chemical Physics of Ice ͑Cambridge University Press, Cambridge, England, 1970͒. proton moves that are large enough to reach across the simu- 3 ϫ ϫ D. Eisenberg and W. Kauzmann, The Structure and Properties of Water lation box. The 4 2 2 box with 128 molecules is 8 mol- ͑Oxford University Press, New York, 1969͒. ecules wide in each direction. Adding a unit cell in each 4 M. Tyagi and S. S. N. Murthy, J. Phys. Chem. A 106, 5072 ͑2002͒. direction increases the width by two molecules in the x di- 5 A. Rahman and F. H. Stillinger, J. Chem. Phys. 57, 4009 ͑1972͒. rection and by four in the y and z directions. To sample 6 A. Yanagawa and J. F. Nagle, Chem. 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At low temperatures, around 50 K, there is 28 H. E. Alper and R. M. Levy, J. Chem. Phys. 91, 1242 ͑1989͒. 29 ͑ ͒ evidence of a transition to a proton-ordered phase. An exami- Y. Guissani and B. Guillot, J. Chem. Phys. 98, 8221 1993 . 30 S. W. Rick, J. Phys. Chem. 114, 2276 ͑2001͒. nation of proton order parameters indicates that the disor- 31 L. Onsager and M. Dupuis, in Electrolytes, edited by B. Pesce ͑Pergamon, dered phase is not completely random with a preference for Oxford, 1962͒,p.27. the energetically lower water orientations. 32 E. R. Batista, S. S. Xantheas, and H. Jońsson, J. Chem. Phys. 109, 4546 ͑1998͒. 33 ACKNOWLEDGMENTS E. R. Batista, S. S. Xantheas, and H. Jońsson, J. Chem. Phys. 112, 3285 ͑2000͒. One of the authors ͑S.W.R.͒ gratefully acknowledges 34 L. Delle Site, A. Alavi, and R. M. Lynden-Bell, Mol. Phys. 96, 1683 support from the Louisiana Board of Regents Support Fund ͑1999͒. 35 ͑ ͒ under Contract No. LEQSF͑2001-04͒-RD-A-42 and the Na- J. F. Nagle, J. Chem. 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