arXiv:1406.5929v1 [physics.chem-ph] 23 Jun 2014 ulhdoe iodrcmail ihteierls(oc- rules ice H-sites the of with posi- cupancy compatible lattice disorder their hydrogen full of in- occupancy ( as fractional tions distribution, a disordered by a dicated present may atoms gen ceitc fteecnesdpae tes-ald‘water so-called (the anomalies’). phases between char- condensed peculiar bonds these some of hydrogen to acteristics rise of give is which presence This molecules, the adjacent understanding. to full some a due water, experimental lack largely of still of phases properties amount solid their the wide of on the works decades. theoretical of few and last spite the in over investigation However, of has diagram subject phase a Ih. pressure-temperature stabil- been the and ice in structures of hexagonal range crystal ity packing usual their of the denser determination in a The application than produces by molecules which obtained water pressures, them high of have of most polymorphs ice identified, crystalline been different sixteen now to ewr ikdb -od.I uhantokec H each a network forming a such entities In defined H-bonds. well by linked as network appear molecules ter hr utb xcl n tmbtentocontigu- two between atom that H and oxygens. one atoms ous two exactly oxygen its be adjacent that must toward there so oriented point is atoms molecule H These each rules. that ice state com- Bernal-Fowler rules disposition so-called a the in with others patible four to H-bonded is molecule nsvrliepae.Oye tm hwfl occupancy full show atoms Oxygen ( phases. ice several in rules.’ f ftercytlorpi ie ( sites crystallographic their of ) ae rsnsalrevreyo oi hss Up phases. solid of variety large a presents Water ntekonieplmrh laigotieX,wa- X), ice out (leaving polymorphs ice known the In rettoa iodro h ae oeue spresent is molecules water the of disorder Orientational < f ) o xml,hxgnlieI presents Ih ice hexagonal example, For 1). 1,4,5 ii.Ti ehdtrsott epeieeog ogv reli give to enough precise be to out turns entropy s method a high This from on system based the limit. drives procedure path Carlo thermodynamic H- employed Monte the totally A a rules. between ice path Bernal-Fowler a along integration thermodynamic ASnmes 65.40.gd,61.50.Ah,64.70.kt numbers: PACS cons connective networks. particu the ice A and the size. entropy on compa ring bet configurational by mean 3.3% the the rationalized of as between been highest such difference has The polymorphs, a structures various XII. with the ice and respectively, the VI, XII, on a V, and entropy two-d IV, H-disordered VI a III, for ices on II, given to model Ic, ice are Ih, ices the entropy including for configurational checked the is method for the of precision 6 ewl ee oteerlssml s‘ice as simply rules these to refer will We h ofiuainletoyo eea -iodrdiepol ice H-disordered several of entropy configurational The ofiuainletoyo yrgndsree c polymo ice hydrogen-disordered of entropy Configurational .INTRODUCTION I. nttt eCecad aeilsd ard osj Super Consejo Madrid, de Materiales de Ciencia de Instituto f s th 0 = fdffrn c hssi h hroyai ii nme of (number limit thermodynamic the in phases ice different of . ) te oyopssc as such polymorphs Other 5). in´ fia CI) apsd atbac,209Mdi,S Madrid, 28049 Cantoblanco, de Campus Cient´ıficas (CSIC), f alsP err n aalRam´ırez Rafael and Herrero P. Carlos ) u hydro- but 1), = Dtd a 0 2018) 20, May (Dated: 1–3 2 O oyops n tpeetsxsc ar fiestruc- ice VII-VIII, ice of VI-XV, V-XIII, XII-XIV, other pairs III-IX, and such for Ih-XI, six known: occurs are present similar tures at for Something and happens XI. polymorphs, as and topology, Ih same finds ices the one Thus, sharing topology. network polymorphs their to according fied ewrs u hyaenteuvln otentokof network sub- the polymorph. disjoint ice to two known equivalent of other not up for any are VI-XV, made they pair also but the networks, is case, network ice another the the to is which equivalent There them two of network). (each of Ic other consists each X) ice to and bonded the hydrogen- VIII not topo- that ices subnetworks nine interpenetrating note as independent has well We (as one network Then, structures. VII ice entity. own different their logically molecules fact water point in that middle so lose that the atoms, on difference oxygen situated main adjacent are the between atoms with hydrogen X VIII, equiv- ice and topologically in VII ice is ices and X to (H-ordered), ice alent Finally, II ice (H-disordered). been (H-disordered), IV has pair Ic no ice which for found: order/disorder structures ice an polymorphs, other twelve these through exist to there other addition In the and transition. to topology phase network one case, same related each the are In share H-ordered. polymorphs is both second the and H-disordered c IaeHodrd hl oepae sie I n V and III occupan- fractional ices cies (some as order phases hydrogen some partial while display H-ordered, are II ice soitdt iodri c a rsne yPauling, by presented found was ice who in disorder H to associated oeue.Ti euttre u ob nrte good Ih, rather ice the of in from capacity obtained be heat entropy experimental to ‘residual’ out the turned with agreement result This molecules. aclto i o osdrteata tutr fthis of structure actual the consider not did calculation h ite nw rsaln c hsscnb classi- be can phases ice crystalline known sixteen The h rteauto ftecngrtoa entropy configurational the of evaluation first The f 0 6= n oetetoyvle correspond values entropy lowest and atdrvdfo efaodn walks self-avoiding from derived tant iodrdsaeadoeflligthe fulfilling one and state disordered al odcreainhsbe found been has correlation good larly rneet nsvrlpolymorphs, several on rrangements mrh scluae ymaso a of means by calculated is ymorphs igi ihsrcua aaeesof parameters structural with it ring . eprtrst h low-temperature the to temperatures ente.Tedpnec fthe of dependence The them. ween 5). ml nrymdli sd othat so used, is model energy imple bevle o h configurational the for values able S mninlsur atc.Results lattice. square imensional 7 = hr h rtplmrhi ahpi is pair each in polymorph first the where o eInvestigaciones de ior Nk B molecules ln(3 pain / )fracytlof crystal a for 2) N ∞ → rphs 9,10 .The ). vntog the though even N water S 8 2 ice polymorph. Nagle11 calculated later the residual en- atoms, forming a hydrogen bond, and second, each oxy- tropy of hexagonal ice Ih and cubic ice Ic by a series gen atom is surrounded by four H atoms, two of them method, and found for both polymorphs very similar val- covalently bonded and two other H-bonded to it. Fulfill- ues, which were close to but slightly higher than the Paul- ment of these rules implies that short-range order must ing’s estimate. In recent years, Berg et al.12,13 have em- be present in the distribution of H atoms in ice poly- ployed multicanonical simulations to calculate the con- morphs. figurational entropy of ice Ih, assuming a disordered H Calculating the configurational entropy of different ice distribution consistent with the ice rules. Furthermore, structures, assuming that the hydrogen distribution is the configurational entropy of partially ordered ice phases only conditioned by the ice rules, is equivalent to find has been studied earlier,14–16 and will not be addressed the number of possible hydrogen configurations compati- here. ble with these rules. Since the entropy is a state function, Besides their application to condensed phases of wa- one can employ any thermodynamic integration that con- ter, ice-type models are relevant for other types of solids nects the required state with any other state whose ther- displaying atomic or spin disorder,17,18 as well as in modynamic properties are known. For the present prob- statistical-mechanics studies of lattice problems.19,20 At lem, this can be adequately done by considering a simple present, no analytical solutions for the ice model in actual model, in which an energy function is defined so that all three-dimensional (3D) ice structures are known, so that states compatible with the ice rules are equally probable no exact values for the configurational entropy are avail- (have the same energy U0), and other hydrogen configu- able. An exact solution was found by Lieb21,22 for the rations have higher energy. The calculation of the con- two-dimensional (2D) square lattice, where the entropy figurational entropy of ice is thus reduced to finding the 3/2 resulted to be S = NkB ln W , with W = (4/3) = degeneracy of the ground state of considered model. Tak- 1.5396, a value somewhat higher than Pauling’s result ing the H configurations according to their Boltzmann WP =1.5. weight, as is usual in the canonical ensemble, only those A goal of the present work consists in finding relations with the minimum energy U0 will appear for T → 0. between thermodynamic properties and topological char- In this respect, the only requirement for a valid energy acteristics of ice polymorphs. In this line, it has recently model is that it has to reproduce the ice rules at low been shown that the network topology plays a role to temperature. Thus, irrespective of its simplicity, it will quantitatively describe the configurational entropy asso- yield the actual configurational entropy if an adequate ciated to hydrogen disorder in ice structures.23 This has thermodynamic integration is carried out. been known after the work by Nagle,11 who showed that Our model is defined as follows. We consider an ice the actual structure is relevant for this purpose, by com- structure as defined by the positions of the oxygen atoms, paring his results for ices Ih and Ic with Pauling’s value so that each O atom has four nearest O atoms. This for the configurational entropy, which is in fact the lim- defines a network, where the nodes are the O sites, and iting value corresponding to a loop-free network (Bethe the links are H-bonds between nearest neighbors. The lattice). network coordination is four, which gives a total of 2N In this paper, we present a simple but ‘formally exact’ links, N being the number of nodes. We assume that on numerical method, to obtain the configurational entropy each link there is one H atom, which can move between of H-disordered ice structures. It is based on a thermo- two positions on the link (close to one oxygen or close dynamic integration from high to low temperatures for to the other). We will use reduced variables, so that a model whose lowest-energy states are consistent with quantities such as the energy U and the temperature T the ice rules. The configurational space is sampled by are dimensionless. The entropy per site s presented below means of Monte Carlo (MC) simulations, which provide is related with the physical configurational entropy S as us with accurate values for the heat capacity of the con- S = NkBs, kB being Boltzmann’s constant. sidered model. This method has been employed earlier For a given configuration of H atoms on an ice network, to calculate the entropy of H-disordered ices Ih and VI in the energy U is defined as: Ref. 23. Apart from ice Ih, Ic, and VI, we are not aware N of any calculation of the configurational entropy of other U = E(in) (1) H-disordered ice polymorphs. Entropy calculations have nX=1 been also carried out earlier for H distributions displaying where the sum runs over the N nodes in the network, and partial ordering on the available crystallographic sites, as i is the number of hydrogen atoms covalently bonded to happens for ices III and V.15 n the oxygen on site n, which can take the values in = 0, 1, 2, 3, or 4. The energy associated to site n is defined as E(in) = |in − 2| (see Fig. 1), having a minimum for II. METHOD OF CALCULATION in = 2 (E2 = 0), which imposes accomplishment of the ice rules at low temperature. In this way, all hydrogen For concreteness, we recall the ice rules, as established configurations compatible with the ice rules on a given by Bernal and Fowler6 in 1933: First, there is one hy- structure are equally probable, because they have the drogen atom between each pair of neighboring oxygen same energy (U0 = 0 in our case). 3

i 0 1 2 3 4 g(i) 1 4 6 4 1 E(i) 2 1 0 1 2

FIG. 1: Schematic illustration of the different hydrogen configurations around an oxygen atom. Open and solid circles represent oxygen and hydrogen atoms, respectively. For each configuration, i is the number of H atoms close (covalently bonded) to the central oxygen, g(i) indicates its multiplicity, and E(i) is its dimensionless energy in the model described in the text.

This energy model is a convenient tool to derive the Here, 2N is the number of links in the considered net- configurational entropy, but it does not represent any re- work, and 22N is the total number of possible configura- alistic interatomic interaction. In fact, we are not dealing tions in our model, as two different positions are accessi- with an actual ordering process in an H-bond network, ble for an H atom on each link. but with a numerical approach to ‘count’ H-disordered Although Eq. (2) is exact in principle, a practical prob- configurations. One can obtain the number of configura- lem in this thermodynamic integration appears because tions compatible with the ice rules on a given structure the limit T → ∞ cannot be reached in actual simula- by a thermodynamic integration, going from a reference tions. One can introduce a high-temperature cutoff Tc state (T → ∞) for which the H configuration is random for this integration, but even if Tc is very high (in our (does not follow the ice rules), to a state in which these units Tc ≫ 1), it may introduce uncontrolled errors in rules are strictly satisfied (T → 0). the calculated entropies. Thus, a reasonable solution to This model assumes that all hydrogen arrangements this problem can be calculating the integral in Eq. (2) that satisfy the Bernal-Fowler rules are energetically de- up to a temperature Tc, and then obtaining the remain- generate. In the actual ice structures there will be vari- ing part (from Tc to T = ∞) by an alternative method. ations in the energy due to different hydrogen bonding With this purpose, an analytical procedure has been in- patterns.24,25 For large system size and at temperatures troduced in Ref. 23, and is described in the following at which the H-disordered phases are stable, the crystal section dealing with an independent-site model. structures corresponding to different hydrogen arrange- ments will form a nearly degenerate energy band, pop- ulated almost uniformly. Thus, the idea underlying the A. Independent-site model calculation of ‘residual’ entropies is that the accessible hydrogen configurations do not correspond to the actual For a real ice network, thermodynamic variables at stable phase at T = 0 (which should be ordered), but high temperatures (T ≫ 1 here) can be well described to an equilibrium H-disordered phase at higher T . This by considering ‘independent’ nodes, as in Pauling’s orig- means that small energy differences between different hy- inal calculation for a hypothetical loop-free network. In drogen arrangements are supposed to not affect signifi- such a case, the partition function can be written as cantly their relative weight for the residual entropy, as zN these arrangements will have nearly the same probabil- Z = , (5) ity in the (equilibrium) disordered phase. 22N We obtain the configurational entropy per site at tem- z being the one-site partition function, and the term 22N perature T from the heat capacity, cv, as in the denominator appears to avoid counting links with T ′ zero or two H atoms. This correction is in fact similar to cv(T ) ′ 8 s(T )= s(∞)+ ′ dT (2) that introduced by Pauling in his entropy calculation. Z∞ T To obtain the partition function z, we consider the 16 where possible configurations of four hydrogen atoms, as given 1 dhUi in Fig. 1, which yields at temperature T : cv(T )= , (3) N dT 4 E(i) − − z = g(i) exp − =6+8e 1/T +2e 2/T . (6) and the entropy in the high-temperature limit (T → ∞)  T  is given by Xi=0 1 From this expression for z one obtains the partition func- s(∞)= ln(22N ) = 2 ln 2 (4) N tion Z using Eq. (5). Thus, the high-temperature limit 4

2 of Z is Z∞ = 16N /4N = 2 N , which is the number of TABLE I: Crystal system and space group for the ice configurations compatible with the condition of having polymorphs studied in this work, along with the references an H atom per link, as indicated above. from where crystallographic data were taken. Nmax is the In this independent-site model, the energy per site at number of water molecules in the largest supercell employed temperature T is here for each ice structure.

1 2 ∂ ln Z 4 −1 −2 hui = T = 2e /T +e /T . (7) Phase Crystal system Space group Ref. Nmax N ∂T z   Ih Hexagonal P 63/mmc, 194 48 2880 For the entropy per site we have Ic Cubic Fd3¯m, 227 49 2744 1 ∂ (T ln Z) II Rhombohedral R3,¯ 148 50 2592 s = , (8) N ∂T III Tetragonal P 41212, 92 51 2592 IV Rhombohedral R3¯c, 167 52 2400 and using Eqs. (5) and (7), one finds V Monoclinic A2/a, 15 51 2688 hui VI Tetragonal P 42/nmc, 137 53 3430 s = ln z + − 2 ln 2 . (9) T XII Tetragonal I42¯ d, 122 54 2400 In the limit T → 0 (ice rules strictly fulfilled), we have

hui Crystallographic data employed to generate the ice struc- lim =0 , (10) T →0 T tures are given in Table I, along with the corresponding references from where they were taken. In this Table we 3 and z → 6 [see Eq.(6)], so that s(0) = ln 2 , which coin- also present the size Nmax of the largest supercells em- cides with Pauling’s result.8 ployed for the different ice polymorphs. Details of the For our calculations on real ice networks we will employ simulations are similar to those employed earlier for ices 23 a cutoff temperature Tc = 10. Introducing this temper- Ih and VI. For the hydrogen distribution on the avail- ature in Eq. (9), we find for the independent-site model able sites along the MC simulations, we assumed peri- s(Tc) = 1.38414, which coincides with the value obtained odic boundary conditions. Sampling of the configuration in Ref. 23 from a series expansion in powers of 1/T up space at different temperatures was carried out by the to fourth order. Metropolis update algorithm.26 For each network we con- sidered 360 temperatures in the range between T = 10 and T = 1, and 200 temperatures in the interval from T = B. Actual ice networks 1 to T = 0.01. For each considered temperature, we car- ried out 104 MC steps for system equilibration, followed The above independent-site equations are exact for a by 8×106 steps for averaging of thermodynamic variables. loop-free network, i.e., the so-called Bethe lattice or Cay- A MC step included an update of 2N (the number of ley tree.19,20 For real ice structures, the corresponding H-bonds) hydrogen positions successively and randomly networks contain loops, which means that the factoriza- chosen. Once calculated the heat capacity cv(T ) at the tion of the partition function in Eq. (5) is not possible. considered temperatures, the integral in Eq. (11) was nu- Nevertheless, at high temperatures thermodynamic vari- merically evaluated by using Simpson’s rule. Finite-size ables for the real structures converge to those of the scaling was employed to obtain the configurational en- independent-site model, and this is the fact we use to tropy per site sth corresponding to the thermodynamic obtain the high-temperature part of our thermodynamic limit (extrapolation to infinite size, N → ∞). integration. Although the heat capacity cv(T ) can be obtained from Thus, for the actual ice networks we calculate the con- MC simulations by numerical differentiation using di- figurational entropy per site for hydrogen-disordered dis- rectly its definition in Eq. (3), we found more practical tributions compatible with the ice rules (accomplished in to derive it from the energy fluctuations at temperature our method for T → 0) as: T . Thus, we employed the expression27

Tc ′ 2 cv(T ) ′ (∆U) s(0) = s(Tc) − ′ dT (11) cv(T )= 2 , (12) Z0 T NT 2 2 2 where s(Tc) = 1.38414 is the value obtained for the where (∆U) = hU i − hUi . In particular, using this independent-site model, and the heat capacity cv is de- expression we obtained cv values with smaller statistical rived from MC simulations for the considered ice poly- noise than employing Eq. (3). morph in the temperature range from T = 0 to T = Tc Now a relevant question is whether the high- (we will take Tc = 10 here). temperature replacement of real ice structures by the Using the energy model described above, we carried independent-site model introduces any appreciable er- out MC simulations on ice networks of different sizes. ror in the calculated configurational entropies. We have 5

0.25

Analytical Analytical formula 0.6 MC ice Ih 0.2 Monte Carlo, ice Ih 0.5

0.15 0.4

0.3 0.16

Heat capacity 0.1 Heat capacity 0.12 0.2 0.08

0.04 0.1 0.05 0.08 0.12 0.16 0 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 Temperature Temperature

FIG. 2: Heat capacity per site as a function of temperature. FIG. 3: Heat capacity per site as a function of tempera- Solid circles are data points obtained from MC simulations ture. Solid circles are data points obtained from simulations for ice Ih. Error bars are less than the symbol size. The for the structure of ice Ih. Error bars are less than the sym- solid line represents the analytical function obtained for the bol size. The solid line corresponds to the independent-site independent-site model as cv(T )= dhui/dT with hui given by model: cv(T )= dhui/dT with hui given by Eq. (7). Eq. (7).

happens at T < 0.4. We observe in the inset that an checked this point in two different ways. First, we have appreciable difference appears in the region between T = compared for some temperatures higher than Tc the en- 0.10 and 0.14. Note that the area under the curve cv(T ) ergy and heat capacity obtained in the independent-site is independent of the considered network, since it is given model with those derived from MC simulations for actual by the difference between the high- and low-temperature ice structures. Second, we have calculated the configura- limits of the energy: tional entropy obtained with our procedure in the low-T ∞ U0 limit for the ice model on a 2D square lattice, for which c (T ) dT = lim hui− =0.75 (13) v →∞ an exact analytical solution is known.21 Both procedures Z0 T N indicate that the error of our method is smaller than the For the energy model considered here we have U0 = 0. error bars associated to the statistical uncertainty of the An alternative procedure to calculate the configura- MC simulations and the error due to the numerical inte- tional entropy can consist in directly obtaining the den- gration of the ratio cv/T in Eq. (11). This is discussed sity of states as a function of the energy, as described else- below. where for order/disorder problems in condensed-matter 28 In Fig. 2 we present the heat capacity cv(T ) in the tem- problems. Thus, the entropy can be obtained in the perature region from T = 1 to 3.5. Symbols are results present problem from the number of states with U = 0, of our MC simulations for ice Ih, whereas the solid line i.e., compatible with the ice rules. is the analytical result obtained for the independent-site To avoid any possible misunderstanding, we note that model as cv(T ) = dhui/dT , with hui given in Eq. (7). the simple model used in our calculations is not aimed at Data obtained by both procedures coincide within the reproducing any physical characteristic of ice polymorphs statistical error bars of the simulation results. Although (as order/disorder transitions) beyond calculating the not appreciable in the figure, we observe a trend of the cv entropy of H distributions compatible with the Bernal- values derived from MC simulations to be slightly larger Fowler rules. This model implicitly takes for granted that than the analytical ones at temperatures T ∼ 1. At tem- the distribution of H atoms on an ice network does not peratures T ∼ Tc, differences between both methods are necessarily display long-range order, and only requires clearly less than the error bars. strict fulfillment of the ice rules (short-range order) for The heat capacity reaches a maximum at Tmax ∼ 0.4, T → 0. The configurational entropy derived in this way and with our numerical precision vanishes at tempera- is what has been traditionally called residual entropy of tures T < 0.08. Larger differences between both proce- ice.8 Taking all this into account, we are not allowing for dures are observed at temperatures T < 1, as shown in any violation of the third law of Thermodynamics, which Fig. 3. At T & 0.4, results of the MC simulations are implies that for T → 0 an ice polymorph displaying H slightly larger than the analytical ones, and the opposite disorder cannot be the thermodynamically stable phase. 6

Thus, it is generally accepted that the equilibrium ice polymorphs in the low-temperature limit show ordered 0.436 proton structures, as expected from a vanishing of the entropy. In this context, order-disorder transitions have 2D square lattice 2,3,25 been observed between several pairs of ice phases. 0.435 These transitions are accompanied by an orientational reorganization of water molecules, which turns out to be a kinetically unfavorable rearrangement of the H-bond 0.434 network. slope 1.05 0.433

III. RESULTS Entropy per site

We have applied the integration method described 0.432 above to calculate the configurational entropy of the ice sth = 0.43152 (exact) model in eight structures. These structures include the 0.431 ice polymorphs showing hydrogen disorder, but also the 0 0.001 0.002 0.003 0.004 network of ice II, for which no hydrogen disorder has been 1 / N observed. In this case, the results presented below have to be understood to apply to a hypothetical structure FIG. 4: Entropy per site sN vs inverse lattice size (1/N) with the same topology as ice II. Moreover, for ices III for the 2D square lattice. Solid squares indicate results of our and V, a partial hydrogen disorder has been found (site thermodynamic integration in the limit T → 0. Error bars are occupancies f different from 0.5), a question that will in the order of the symbol size. A solid circle shows the exact not be taken into account here (see the discussion below analytical result obtained by Lieb in the limit N →∞.21 on this question). Ice VII will not be discussed, since it consists of two sub-networks, each of them equivalent to the ice Ic network (see the Introduction). for ices IV, V, and VI we find α = 2.00(2), 1.91(3), and For each considered network, we obtained the configu- 3.44(6), respectively. For comparison, arrows in Fig. 5 rational entropy sN in the limit T → 0 for several super- indicate the entropy per site for the 2D square lattice cells of size N, as described in Sect. II. Then, the entropy and the Pauling result. The data for the ice polymorphs per site in the thermodynamic limit, sth, is obtained by converge in the infinite-size limit to sth between those extrapolating N → ∞. The precision of our procedure two values. can be checked by calculating the configurational entropy For other ice structures not shown in Fig. 5 we also for the ice model on a 2D square lattice, for which an ex- 21,22 obtained sth values between the square lattice and the act analytical solution is known. Pauling result (see Table II). The largest value corre- In Fig. 4 we present s for the 2D square lattice as a N sponds to ice VI (sth = 0.42138(11)), and the smallest function of the inverse supercell size 1/N. Solid squares one was found for ice XII (sth = 0.40813(5)). Between represent the entropy values yielded by our thermody- both values one finds a relative difference of about 3%. namic integration. It is found that the entropy per site For hexagonal ice Ih, the configurational entropy is found sN decreases for increasing size N, and a linear depen- to be an 1.3% larger than Pauling’s estimate. For other dence of s on 1/N is observed for N & 150, as N ice polymorphs, the difference between sth and the Paul- α ing result ranges from 0.7% for ice XII to 3.9% for ice sN = sth + , (14) VI. N Information on the least-square fit of sN values for with a slope α =1.05. For supercell sizes N < 150 (not different ice polymorphs, such as the parameter α, the displayed in Fig. 4), sN values slightly deviate from the correlation coefficient ρ, and the number of points np linear trend given in Eq. (14), being smaller than those employed in the fit, are given in Table II. When dis- predicted from the fit for N > 150. Extrapolating sN to cussing the configurational entropy of ice, it is usually infinite size (1/N → 0) gives a value sth = 0.43153(3), in given in the literature the parameter W , instead of the agreement with the exact solution for the square lattice entropy sth itself. Here, we find directly sth from ther- 21 derived by Lieb from a transfer-matrix method: sth = modynamic integration and finite-size scaling, so that W 3 2 ln(4/3)=0.43152. values given in Table II were calculated for the different For the 3D structures of ice polymorphs we also found ice polymorphs as W = exp(sth). The error bars ∆W a linear dependence of the configurational entropy sN on and ∆sth for the values of sth and W are related by the the inverse supercell size. This is shown in Fig. 5 for ices expression ∆W = W ∆sth, as can be derived by differ- IV, V, and VI. One observes that both the slope α and entiating the exponential function. Error bars for sth the infinite-size limit sth depend on the ice structure. The represent one standard deviation, as given by the least- slope is clearly larger than in the 2D case (α =1.05), as square procedure employed to fit the data. These error 7

TABLE II: Configurational entropy sth associated to hydrogen disorder in several ice polymorphs, as derived from our Monte Carlo simulations. For comparison we also give results for the 2D square lattice, as well as Pauling’s value (Bethe lattice). W = exp(sth); np: number of data points employed in the linear fits; α: slope of the linear fit as in Eq. (14); ρ: correlation coefficient.

Ice np α sth W ρ

Ih 9 1.84(2) 0.41069(8) 1.50786(12) 0.9994

Ic 10 1.82(2) 0.41064(7) 1.50778(11) 0.9994

II 12 1.87(1) 0.40919(4) 1.50560(6) 0.9998

III 12 1.79(2) 0.41165(7) 1.50931(11) 0.9990

IV 10 2.00(2) 0.40847(4) 1.50451(6) 0.9997

V 8 1.91(3) 0.41316(6) 1.51159(9) 0.9992

VI 9 3.44(6) 0.42138(11) 1.52406(16) 0.9990

XII 10 2.08(3) 0.40813(5) 1.50400(8) 0.9992

Square 12 1.05(1) 0.43153(3) 1.53961(5) 0.9994

Pauling - - 0.405465 1.5 -

0.44 tice (linear fit shown in Fig. 4) and for a loop-free network with coordination z = 4 (Pauling’s result). Note that, al- though the result obtained by Pauling was intended to 2D square VI reproduce the residual entropy of ice Ih, it does not take 0.43 into account the actual ice network, but only the fourfold coordination of the structure. V Now the question arises why the entropy per site de- creases for increasing supercell size. A qualitative expla- 0.42 nation is the following. Let us call ΩN the number of configurations compatible with the ice rules for supercell IV size N. For two independent cells of size N, the number

Entropy per site 2 of possible configurations is then ΩN . If one puts both 0.41 cells together to form a larger cell of size 2N, one has Ω < Ω2 , since configurations not fitting correctly at Pauling 2N N the border between both N-size cells are unsuitable and have to be rejected. Then, one has 0.4 0 0.001 0.002 0.003 0.004 0.005 1 1 2 1 1 / N s2 = ln Ω2 < ln Ω = ln Ω = s (15) N 2N N 2N N N N N

FIG. 5: Entropy per site sN as a function of the inverse Something similar can be argued in general for two differ- supercell size (1/N) for three ice structures: IV, V, and VI. ent cell sizes, N

−4 fact, this is what we found from our simulations for the bars are less than ±10 , except for ice VI, for which we different ice polymorphs, with the parameter β so small found ∆s =1.1 × 10−4. th that a linear dependence of sN on 1/N is consistent with For comparison with the results found for the ice struc- the results of the thermodynamic integration at least for tures, we also give in Table II data for the 2D square lat- N > 150. 8

29–31 For ice Ih we find sth = 0.41069(8). As already ob- study properties of various types of materials. For served in the analytical result by Nagle11 and in the crystalline ice polymorphs, a discussion of different net- multicanonical simulations by Berg et al.,12,13 the con- work topologies and the relation of ring sizes in the var- figurational entropy is higher than the earlier estimate ious phases with the crystal volume was presented by by Pauling. Our result is somewhat higher than that of Salzmann et al.7, and more recently by the present au- 11 23 Nagle, who found sth = 0.41002(10), and slightly higher thors in Ref. 32. As indicated earlier, one expects that than the results of multicanonical simulations, although structural and topological aspects of ice networks can the latter may be compatible with our data, taking into be relevant to understand quantitatively the configura- account the error bars. We do not find at present a clear tional entropy of H-disordered structures. In particular, reason why our result for ice Ih is higher than that found this entropy should in some way be related with effective by Nagle using a series method. The error bar given by parameters describing topological aspects of the corre- this author is not statistical, and one could argue that sponding networks. maybe it was underestimated, but this question should be further investigated. A discussion on the various re- TABLE III: Minimum (Lmin), maximum (Lmax), and mean sults for ice Ih is given in Ref. 23. ring size hLi for different ice structures, along with the cor- For cubic ice Ic, we found sth = 0.41064(7), which co- responding parameter a and connective constant µ. a is the incides within error bars with our result for hexagonal ice coefficient of the quadratic term in the coordination sequence, 2 Ih. To our knowledge, the only earlier calculation for ice Mk ∼ ak , as in Eq. (16). Ic is that presented by Nagle in 1966,11 who concluded that within the limits of his estimated error, the entropy Network Lmin Lmax hLi a µ is the same for ices Ic and Ih, i.e., sth = 0.41002(10). We reach the same conclusion concerning both ice poly- Ih 6 6 6 2.62 2.8793 morphs, but in our case the entropy value is higher by an 0.15%. This difference seems to be small, but it is Ic 6 6 6 2.50 2.8792 significant as it amounts to more than eight times our II 6 10 8.52 3.50 2.9049 error bar (standard deviation). It is known that ices III and V show partially ordered III 5 8 6.67 3.24 2.8714 hydrogen distributions. This means that some fractional occupancies of H-sites are different from 0.5. Then, the IV 6 10 9.04 4.12 2.9162 actual configurational entropy for these polymorphs is V 4 12 8.36 3.86 2.8596 lower than that corresponding to a H-disordered distri- bution compatible with the ice rules. MacDowell et al.15 VI 4 8 6.57 2.00 2.7706 calculated the configurational entropy of partially or- XII 7 8 7.6 4.26 2.9179 dered ice phases by using an analytical procedure, based on a combinatorial calculation. For ices III and V, these Square 4 4 4 0.0 2.6382 authors found entropy values sth = 0.3686 and 0.3817, respectively, which they used to calculate the phase dia- Bethe – – – – 3.0000 gram of water with the TIP4P model potential. Taking the Pauling result as the entropy value for H-disordered ice phases, the data by MacDowell et al. mean an entropy The tendency of the entropy sth to increase due to the reduction of 9.1% and 5.9%, for ice III and V, respec- presence of loops in the ice structures has been pointed tively. This is an appreciable entropy reduction, which is 23 15 out recently. The Pauling approximation ignores the noticeable in the calculated phase diagram. However, presence of loops, as happens in the Bethe lattice,19,20 if one takes for the H-disordered phases more realistic and yields a value sth =0.40547. For ice Ih, which con- entropy values, as those derived here, the entropy drop tains six-membered rings of water molecules, the config- due to partial hydrogen ordering is somewhat larger. In urational entropy is higher than Pauling’s value by an fact, one finds 10.5% for ice III and 7.6% for ice V. In 1.3%, and it is still higher for the ice model on a 2D any case, it is known that the hydrogen occupancies of square lattice, including four-membered rings (a 6.4% the different crystallographic sites may change to some with respect to the Pauling approach). For other ice poly- extent with the temperature, so that the configurational 15 morphs with larger ring sizes, one can expect a smaller entropy will also change. configurational entropy for disordered hydrogen distri- butions, and thus closer to the Pauling result. In Ta- ble III we give the minimum (Lmin), maximum (Lmax), IV. RELATION WITH STRUCTURAL and mean ring size hLi for the ice structures considered PROPERTIES here. In this line, we consider the mean loop size hLi as Topological characteristics of solids at the atomic or a first topological candidate to be compared with sth. molecular level have been considered along the years to A correlation between both quantities was suggested in 9

0.44 greatly affects the actual value of the entropy sth. Thus, the configurational entropy for ice VI turns out to be clearly higher than that of ice III, even though both poly- 2D square morphs have similar values of hLi. However, the effect of 0.43 four-membered rings is smaller for larger hLi, as can be seen by comparing sth values for ices II an V (see Fig. 6). VI Further than structural rings, the topology of a given ice network can be characterized by the so-called coordi- 0.42 nation sequences {Mk} (k = 1, 2, ...), where Mk is the number of sites at a topological distance k from a refer- V III ence site (we call topological distance between two sites

Entropy per site the number of bonds in the shortest path connecting one 0.41 IV site to the other).32 For 3D structures, M increases at Ih, Ic k large distances as: XII II 2 Mk ∼ a k , (16) 0.4 3 4 5 6 7 8 9 10 where a is a network-dependent parameter. Mk increases Mean loop size quadratically with k just as the surface of a sphere in- creases quadratically with its radius. For structures in-

FIG. 6: Configurational entropy per site sth vs mean loop cluding topologically non-equivalent sites, the actual co- size hLi for several ice structures. Error bars for the entropy ordination sequences corresponding to different sites may are less than the symbol size. The 2D square lattice is in- be different, but the coefficient a coincides for all sites in a cluded for comparison. The dashed line is a least-square lin- given structure.32 This parameter a can be used to define ear fit to the data of structures containing four-membered a ‘topological density’ ρt for ice polymorphs as ρt = wa, rings (Lmin = 4). The dashed-dotted line is a fit for the ice where w is the number of disconnected subnetworks in structures with Lmin > 5. the considered network. Usually w = 1, but for ice struc- tures including two interpenetrating networks (as ices VI and VII) one has w = 2. The parameter a and the topo- Ref. 23, and will be considered in more detail here. In logical density ρt of crystalline ice structures have been Fig. 6 we present the configurational entropy of various calculated earlier.32 It was found that a ranges from 2.00 ice polymorphs vs hLi. For comparison, we also include (ice VI) to 4.27 (ice XII), as can be seen in Table III for a data point for the 2D square lattice, where all loops the ice polymorphs studied here. have L = 4. There appears a tendency of the configu- In Fig. 7 we display the configurational entropy per site rational entropy sth to decrease for rising hLi, but one sth of ice polymorphs vs the coefficient a derived from the observes a large dispersion of the data points for differ- corresponding coordination sequences. We include also a ent ice polymorphs, indicating a low correlation between point for the 2D square lattice, for which a = 0, as in both quantities. However, a careful observation of the this case the dependence of Mk on the distance k is lin- data displayed in Fig. 6 reveals that points correspond- ear: Mk =4k. We find that sth decreases for increasing ing to the ice structures are roughly aligned, with the parameter a, but there appears an appreciable dispersion exception of ices V and VI, which are located away from in the data points, which can be associated to the size the general trend. A common characteristic of these two of structural rings in the considered ice polymorphs. In ice polymorphs is that they contain four-membered rings particular, ice networks including four-membered rings (Lmin = 4). In fact, we find a good linear correlation be- (Lmin = 4) depart clearly from the general trend for the tween sth and the mean ring size hLi for these structures other structures, similarly to the fact observed above in and the 2D square lattice (dashed line in Fig. 6). The Fig. 6 for the relation between entropy and mean loop dashed-dotted line is a linear fit to the data points of ice size. Thus, two different fits are also displayed in Fig. 7. structures not including four-membered rings. For these The dashed line is a linear fit for the ice polymorphs with structures we have Lmin > 4, i.e. Lmin = 5 for ice III, Lmin = 4 (ices V and VI), along with the 2D square lat- Lmin = 6 for ices Ih, Ic, II, and IV, and Lmin = 7 for tice. The dashed-dotted line is a least-square linear fit ice XII. We observe that the point corresponding to ice for the structures with Lmin > 5, with all of them hav- III appears above the dashed-dotted line, whereas that ing Lmin = 6 except ice XII, for which Lmin = 7. There corresponding to ice XII lies below it. This appears to remains ice III (Lmin = 5), not included in either of the be in line with the trend found for ices V and VI, since fittings, whose data point (solid triangle) lies on the re- Lmin(III) < 6 < Lmin(XII), indicating that for a given gion between both lines in Fig. 7. value of hLi, a lower Lmin causes a larger entropy sth. From these results we conclude that the configurational These observations allow us to conclude that the con- entropy tends to decrease as the parameter a (density of figurational entropy tends to lower as the mean ring nodes) increases. However, the presence of small rings in size increases, but the presence of four-membered rings the ice structures strongly affects the actual value of sth. 10

0.44

2D square 2D square 0.43 0.43 VI VI 0.42 0.42 V V III

Entropy per site Ic Entropy per site 0.41 0.41 IV Ih, Ic Ih II XII Pauling-Bethe 0.4 0.4 0 1 2 3 4 5 2.6 2.7 2.8 2.9 3 Parameter a Connective constant

FIG. 7: Configurational entropy per site sth vs parameter FIG. 8: Configurational entropy per site sth vs connective a of coordination sequences for several ice structures. Error constant µ for several ice structures (open circles). Values for bars for the entropy are less than the symbol size. A point for the 2D square lattice and the Bethe lattice (Pauling model) the 2D square lattice (a = 0) is shown for comparison. The are displayed for comparison (solid squares). The solid line dashed line is a least-square linear fit to the data of structures is a least-square linear fit to the data of ice structures. Error containing four-membered rings (Lmin = 4). The dashed- bars for the entropy are less than the symbol size. dotted line is a fit for the ice structures with Lmin > 5.

µ can be obtained as the limit Thus, sth for ices V and VI (with Lmin = 4) is clearly C larger than that corresponding to ice polymorphs with µ = lim n . (18) →∞ similar a values but with Lmin > 4. n Cn−1 We see that the correlation of the configurational en- tropy with both the mean loop size and the topologi- This parameter µ depends upon the particular topology of each structure, and has been accurately determined cal parameter a is highly affected by the presence of 36 structural rings with small number of water molecules, for standard 3D lattices. For the networks of differ- in particular by those of the smallest size found for the ent ice polymorphs, the connective constant µ has been recently calculated, and was compared with other topo- known crystalline polymorphs, L = 4. It would be de- 35 sirable to find other structural or topological characteris- logical characteristics of these networks. It ranges from tics which could correlate more directly with the entropy 2.770 for ice VI to 2.918 for ice XII, as shown in Table III. In Fig. 8 we present the configurational entropy per sth. In this line, various parameters defined in statistical- mechanics studies of lattice problems have been found site sth vs the connective constant µ for the considered earlier to be useful to characterize several thermody- ice polymorphs, as well as for the loop-free Bethe lat- namic properties. One of these parameters, which can tice (Pauling’s result) and the 2D square lattice. One be used to characterize the topology of ice structures, observes in this figure a negative correlation between the is the ‘connective constant’ µ, or effective coordination entropy sth and µ. The line is a least-square fit to the number.33–35 This network-dependent parameter can be data points of the ice polymorphs. Data corresponding calculated from the long-distance behavior of the number to the square lattice and the Bethe lattice were not in- of possible self-avoiding walks (SAWs) in the correspond- cluded in the fit, as they depart from the trend of the ice ing structures.26,36,37 structures. This seems to be due to the role played by A self-avoiding walk on an ice network is defined as a dimensionality in the actual values of the configurational walk in the simplified structure (only oxygen sites, see entropy (the Bethe lattice has no well-defined dimension Sect. II) which can never intersect itself. The number d, and for some purposes it can be considered as d → ∞). In relation to the data displayed in Fig. 8 for ice poly- Cn of possible SAWs of length n starting from a given network site is asymptotically given by36–38 morphs, we cannot find at present any reason why the correlation between sth and µ should be strictly linear, γ−1 n Cn ∼ n µ , (17) but it is clear that there is a relation between both mag- nitudes. To obtain some insight into this question, a where γ is a critical exponent which takes a value ≈ 7/6 qualitative argument is the following. The connective for 3D structures.36,39,40 Then, the connective constant constant µ is a measure of the mean number of sites 11 connected to a node in long SAWs. Then, for the dis- This model has allowed us to obtain the entropy for dif- tribution of H atoms on the links of the ice structures ferent ices polymorphs, taking into account the actual according to the Bernal-Fowler rules, a larger µ is asso- structure of each of them. This procedure has turned ciated to a rise in the correlations between occupancies out to be very precise, indicating that the associated of different hydrogen sites, as information on the actual error bars can be made relatively small without using occupancy of a link propagates ‘more effectively’ through elaborate procedures, but only employing standard sta- the ice network.41,42 This correlation gives rise to a de- tistical mechanics and numerical techniques. In fact, the crease in the number of configurations compatible with estimated error bars of the calculated entropy values are the ice rules for a given structure, which means a re- about one part in 4,000, i.e. around a 0.03% of the actual duction of the configurational entropy of the hydrogen sth value. distribution on the considered ice network. Similar rela- For real ice structures we found entropy values rang- tions between SAWs and order/disorder problems, such ing from sth = 0.4081 for ice XII to sth = 0.4214 for as Ising or lattice-gas models, have been found and dis- ice VI, which means a difference of 3.3% between these cussed earlier.33,34,43 In this line, the relation observed polymorphs. These values are 0.7% and 3.9% larger than here between configurational entropy and SAWs should Pauling’s result, respectively. For ice Ih we obtained sth be investigated further in the context of the statistical = 0.4107, which is close to the result of earlier calcula- mechanics of lattice models. tions using multicanonical simulations. Going back to the topological characteristics of ice Given the diversity of ice structures, the resulting val- structures, the smallest cycles of water molecules in crys- ues of the configurational entropy have been analyzed in talline ice phases are four-membered rings. As observed view of different aspects of the corresponding networks. in Figs. 6 and 7, and commented above, the presence In this line, it was found a trend of the entropy to lower of these rings in ices V and VI imposes correlations on as the mean ring size hLi increases. However, the pres- the hydrogen atom distribution clearly stronger than in ence of four-membered rings in some polymorphs (ices V other ice phases including larger rings. A qualitative ar- and VI) markedly affects the actual value of the entropy. gument for this fact can be obtained from an estima- A similar conclusion can be derived for the correlation tion of the way in which the presence of rings affects the between the configurational entropy and other topolog- configurational entropy, similarly to calculations carried ical parameters of ice networks, such as the parameter 44 out by Hollins for ice Ih. This author found that the a giving the topological density ρt. A better correla- correction to the Pauling value (W = 1.5) introduced tion has been found between the configurational entropy by six-membered rings (considered to be independent) is of H-disordered ice polymorphs and the corresponding ∆W = 0.0041. Using the same reasoning, with the so- connective constants, derived from self-avoiding walks on called ‘poles’ in Ref. 44, we find for ice VI a correction their networks. This allows one to rationalize the entropy about 4.5 times larger (∆W = 0.0185). These values of differences obtained for the ice polymorphs in terms of ∆W are smaller than those found in the more precise parameters depending on the actual topology of the as- thermodynamic integration given in Table. II, as the for- sociated structures. mer do not take into account correlations such as those The entropy difference found here for real ice poly- present between adjacent rings and at larger distances morphs may be important for detailed calculations of in the network. In any case, it is clear that the correla- the phase diagram of water,45–47 as indicated earlier for tion introduced by four-membered rings is significantly ice phases with partial proton ordering.15 Moreover, the stronger than for larger rings. This stronger correlation method described here to calculate the configurational affects similarly the configurational entropy and connec- entropy of ice can be applied to other condensed-matter tive constant of the H-bond networks studied here, as problems, such as the so-called spin-ice systems, where manifested by the correlation displayed by both quanti- understanding the spin disorder is crucial to explain their ties (see Fig. 8), where all considered ice phases, contain- magnetic properties.17,18 ing or not four-membered rings, follow the same trend.

V. CONCLUDING REMARKS Acknowledgments

We have presented results for the configurational en- This work was supported by Direcci´on General de In- tropy of H-disordered ice structures, as calculated from vestigaci´on (Spain) through Grant FIS2012-31713, and a thermodynamic integration for a simple model which by Comunidad Aut´onoma de Madrid through Program reproduces the Bernal-Fowler rules in the limit T → 0. MODELICO-CM/S2009ESP-1691.

1 V. F. Petrenko and R. W. Whitworth, Physics of Ice (Ox- 2 A. N. Dunaeva, D. V. Antsyshkin, and O. L. Kuskov, Solar ford University Press, New York, 1999). 12

System Research 44, 202 (2010). in Statistical Physics (Springer, Berlin, 2010), 5th ed. 3 T. Bartels-Rausch, V. Bergeron, J. H. E. Cartwright, 27 D. Chandler, Introduction to modern statistical mechanics R. Escribano, J. L. Finney, H. Grothe, P. J. Gutierrez, (Oxford University Press, Oxford, 1987). J. Haapala, W. F. Kuhs, J. B. C. Pettersson, et al., Rev. 28 C. P. Herrero and R. Ram´ırez, Chem. Phys. Lett. 194, 79 Mod. Phys. 84, 885 (2012). (1992). 4 D. Eisenberg and W. Kauzmann, The Structure and Prop- 29 W. B. Pearson, The crystal chemistry and physics of metals erties of Water (Oxford University Press, New York, and alloys (Wiley, New York, 1972). 1969). 30 F. Liebau, Structural chemistry of silicates: structure, 5 G. W. Robinson, S. B. Zhu, S. Singh, and M. W. Evans, bonding, and classification (Springer, Berlin, 1985). Water in Biology, Chemistry and Physics (World Scien- 31 A. F. Wells, Structural inorganic chemistry (Clarendon tific, Singapore, 1996). Press, Oxford, 1986), 5th ed. 6 J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515 32 C. P. Herrero and R. Ram´ırez, Phys. Chem. Chem. Phys. (1933). 15, 16676 (2013). 7 C. G. Salzmann, P. G. Radaelli, B. Slater, and J. L. Finney, 33 C. Domb, J. Phys. C: Solid State Phys. 3, 256 (1970). Phys. Chem. Chem. Phys. 13, 18468 (2011). 34 C. P. Herrero, J. Phys.: Condens. Matter 7, 8897 (1995). 8 L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935). 35 C. P. Herrero, Chem. Phys. 439, 49 (2014). 9 W. F. Giauque and J. W. Stout, J. Amer. Chem. Soc. 58, 36 D. S. McKenzie, Phys. Rep. 27, 35 (1976). 1144 (1936). 37 D. C. Rapaport, J. Phys. A: Math. Gen. 18, 113 (1985). 10 O. Haida, T. Matsuo, H. Suga, and S. Seki, J. Chem. Ther- 38 V. Privman, P. C. Hohenberg, and A. Aharoni, in Phase modyn. 6, 815 (1974). Transitions and Critical Phenomena, edited by C. Domb 11 J. F. Nagle, J. Math. Phys. 7, 1484 (1966). and J. L. Lebowitz (Academic Press, London, 1991), 12 B. A. Berg, C. Muguruma, and Y. Okamoto, Phys. Rev. vol. 14, pp. 1–134. B 75, 092202 (2007). 39 S. Caracciolo, M. S. Causo, and A. Pelissetto, Phys. Rev. 13 B. A. Berg, C. Muguruma, and Y. Okamoto, Mol. Sim. 38, E 57, R1215 (1998). 856 (2012). 40 M. Chen and K. Y. Lin, J. Phys. A: Math. Gen. 35, 1501 14 R. Howe and R. W. Whitworth, J. Chem. Phys. 86, 6443 (2002). (1987). 41 C. P. Herrero, Phys. Rev. E 66, 046126 (2002). 15 L. G. MacDowell, E. Sanz, C. Vega, and J. L. F. Abascal, 42 C. P. Herrero and M. Saboy´a, Phys. Rev. E 68, 026106 J. Chem. Phys. 121, 10145 (2004). (2003). 16 B. A. Berg and W. Yang, J. Chem. Phys. 127, 224502 43 A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002). (2007). 44 G. T. Hollins, Proc. Phys. Soc. 84, 1001 (1964). 17 S. V. Isakov, K. S. Raman, R. Moessner, and S. L. Sondhi, 45 E. Sanz, C. Vega, J. L. F. Abascal, and L. G. MacDowell, Phys. Rev. B 70, 104418 (2004). Phys. Rev. Lett. 92, 255701 (2004). 18 D. Pomaranski, L. R. Yaraskavitch, S. Meng, K. A. Ross, 46 R. Ram´ırez, N. Neuerburg, and C. P. Herrero, J. Chem. H. M. L. Noad, H. A. Dabkowska, B. D. Gaulin, and J. B. Phys. 137, 134503 (2012). Kycia, Nature Phys. 9, 353 (2013). 47 R. Ram´ırez, N. Neuerburg, and C. P. Herrero, J. Chem. 19 J. M. Ziman, Models of disorder (Cambridge University, Phys. 139, 084503 (2013). Cambridge, 1979). 48 S. W. Peterson and H. A. Levy, Acta Cryst. 10, 70 (1957). 20 G. M. Bell and D. A. Lavis, Statistical Mechanics of Lattice 49 H. K¨onig, Z. Kristallogr. 105, 279 (1944). Models. Volume 1: Closed Form and Exact Theories of 50 B. Kamb, W. C. Hamilton, S. J. LaPlaca, and A. Prakash, Cooperative Phenomena (Ellis Horwood Ltd., New York, J. Chem. Phys. 55, 1934 (1971). 1989). 51 C. Lobban, J. L. Finney, and W. F. Kuhs, J. Chem. Phys. 21 E. H. Lieb, Phys. Rev. Lett. 18, 692 (1967). 112, 7169 (2000). 22 E. H. Lieb, Phys. Rev. 162, 162 (1967). 52 H. Engelhardt and B. Kamb, J. Chem. Phys. 75, 5887 23 C. P. Herrero and R. Ram´ırez, Chem. Phys. Lett. 568-569, (1981). 70 (2013). 53 W. F. Kuhs, J. L. Finney, C. Vettier, and D. V. Bliss, J. 24 C. Knight, S. J. Singer, J. L. Kuo, T. K. Hirsch, L. Ojamae, Chem. Phys. 81, 3612 (1984). and M. L. Klein, Phys. Rev. E 75, 056113 (2006). 54 C. Lobban, J. L. Finney, and W. F. Kuhs, Nature 391, 25 S. J. Singer and C. Knight, Adv. Chem. Phys. 147, 1 268 (1998). (2012). 26 K. Binder and D. W. Heermann, Monte Carlo Simulation