CO2 packing polymorphism under pressure: mechanism and thermodynamicsof the I-III polymorphic transition

Ilaria Gimondi1 and Matteo Salvalaglio1, a)

Thomas Young Centre and Department of Chemical Engineering, University College London, London WC1E 7JE,UK.

(Dated: 16 April 2018)

In this work we describe the thermodynamics and mechanism of CO2 polymorphic transitions under pressurefrom form I to form III combining standard molecular dynamics, well-tempered metadynamics and committoranalysis. We find that the phase transformation takes place through a concerted rearrangement of CO2

molecules, which unfolds via an anisotropic expansion of the CO2 supercell. Furthermore, at high pressureswe find that defected form I configurations are thermodynamically more stable with respect to form I withoutstructural defects. Our computational approach shows the capability of simultaneously providing an extensivesampling of the configurational space, estimates of the thermodynamic stability and a suitable description ofa complex, collective polymorphic transition mechanism.

Keywords: polymorphism, carbon dioxide, enhanced sampling, metadynamics, mechanism

I. INTRODUCTION

Polymorphism, namely the possibility that molecularcrystals assemble in the solid phase in different crystallattices, is ubiquitous in nature. The spatial arrange-ment of molecules is key in defining mechanical, physical,chemical, and functional properties of materials. Under-standing the molecular details of the thermodynamicsand mechanisms underlying polymorphism is thereforekey to develop detailed, rational descriptions of manynatural and industrial processes1–8.

In this direction, a notable effort is put in develop-ing both ab initio and enhanced sampling techniques topredict polymorphs of a molecule (in particular, CSPtechniques3,9–11), to evaluate their relative stability at fi-nite temperature and pressure, i.e. at conditions relevantfor the life-cycle of a solid product, and to study transi-tion mechanism and kinetics. Among enhanced samplingtechniques, metadynamics5,12–20 (MetaD) and adiabaticfree energy dynamics21–23 (AFED) are employed in lit-erature to study polymorphism. Indeed, over the yearsthese techniques have been tested, developed and com-pared on benchmark systems and combined with CSPmethods. Such works made a successful step towards thecharacterisation of solid phase transition, proving thesetools to be powerful in the prediction of new structuresand transition pathways as well as of the phase diagramwithout any a priori knowledge.However, a complete and systematic investigation ofpolymorphic transitions is still challenging.

In this work our aim is to exploit state of the art en-hanced sampling simulations to investigate the thermo-dynamics and transition mechanisms at play in polymor-phic transitions. To this aim, we combine well-temperedMetaD and committor analysis in order to identify a

a)Electronic mail: m.salvalaglio@ucl.ac.uk

suitable low dimensional description of the transitionbetween two polymorphic phases in collective variablespace. To do so, here we focus our attention on solidCO2, more precisely on the I-III polymorphic transitionthat characterises CO2 packing polymorphism. Packingpolymorphism arises when two solid phases differ in thepacking of molecules, which have all the same molecularstructure, as opposite to conformational polymorphism2.

In molecular solid phases, CO2 molecules maintaintheir gas phase conformation. To a first, crude, approx-imation, each molecule can in fact be described as rigidand the bending of the O-C-O 180◦ angle can be reason-ably neglected. Thanks to such limited conformationalflexibility, it is as easy as spontaneous to identify eachCO2 molecule with a vector passing through its axis;moreover, the centre of mass corresponds to the carbonatom at every simulation time. As a result, the stateof each molecule can be completely characterised by theposition of its centre of mass and the vector representingits orientation in space.

Despite its simple molecular structure, CO2 has arather complex solid-state phase diagram24 (partially re-ported in Figure 1 (a)). Indeed, at high temperature andpressure, seven different crystal structures have been de-tected so far, among which many are still debated24–42.The first form detected was molecular phase I, also calleddry ice, crystallised directly from the melt; phase III fol-lowed, obtained through the compression of dry ice; thediscovery of a polymeric structure, classified as phase V,attracted more interest to the study of this system, re-sulting in the identification of two more phases, II andIV. Phases II and IV are currently object of discussionas different groups hold contrasting views on their na-ture and role in the transition between molecular tonon-molecular phases24,26,28–30,33,37,43. Furthermore, anamorphous phase (VI) is also identified, and the existenceof molecular form VII as a phase itself is still under in-vestigation (see Figure 1 (a)).

In this work we study the transition between phases I

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FIG. 1. (a) Detail of the phase diagram of CO2 at high temperature and pressure from Datchi at al.24 with highlighted thephases of interest of the present work (I in blue, III in orange). The red dots represent the condition of temperature andpressure investigated. (b) and (d) Snapshots of different planes of the 256-molecule super cell in phase I and III, respectively.(c) Detail of phase III with the typical 52-degree angle φ highlighted; in particular the red arrow aligns with the direction ofthe side of the box, while the black dashed one with the CO2 molecular axis.

and III, which are largely accepted and well characterisedin the literature (Figure 1 (b) and (d)). Their structuralarrangements appear to hold several similarities. Bothpolymorphs, indeed, are face centred with four moleculesin the unit cell, but while polymorph I’s lattice is cubicPa3, III’s is orthorombic Cmca. A major difference is theorientation of the CO2 particles: in phase I the molecularaxis is in fact aligned with the diagonal of the cell, whilein phase III they are arranged in parallel layers in whichmolecules describe a characteristic 52◦ angle, φ, with theside of the lattice (Figure 1 (c)).The I − III transformation takes place at around 11-12(11.8) GPa24 independently of temperature (dPI↔III

dT =0); nevertheless defining the transition conditions is a dif-ficult task, and the pressure transition range is suggestedto be wider (7 - 15 GPa)35, while Olijnyk et al.44 observetransition III to I under unloading at around 2.5 - 4.5GPa at 80 K. There is good agreement on the occurrenceof a hysteresis of the specific volume, which decreases ofabout 2% from I to III36,37. It is also generally acceptedthat the transition takes place through a concerted rota-tion of the molecules together with a deformation of thecubic structure to a parallelepiped, thanks to the pecu-liar geometrical features of the two phases.The early works by Kuchta and Etters38–40 study suchtransition through NPT Monte Carlo (MC) simulationscoupled with equalization of the Gibbs free energy in thephases under investigation, not without uncertainties40.

Moreover, the authors identify the orientation of themolecules in the lattice as the most relevant featurechanging in the transition and thus they employ it as atransition coordinate to estimate the free energy profileassociated to the transformation; their calculations bothat 0 K38 and room temperature40 locate the transitionpressure at 4.3 GPa. Li et al.41 apply instead the sec-ond order Møller Plesset (MP2) technique to the studyof molecular crystals42. Despite small inaccuracies, theyreach a good agreement about the transition pressure(around 11.9 - 12.7 GPa), obtained evaluating the freeenergy of the two polymorphs at different T-P conditions.

Here we aim at complementing the state of the art byproviding an extensive sampling of the configurationalspace explored during the I - III polymorphic transitionwhile contextually identifying the dominant transitionmechanism. In the first part we perform well-temperedmetadynamics simulations45 with two order parametersas collective variables over a range of pressure at 350 K;from these, we obtain free energy surfaces that allow aninsight into the relative stability between phase I andIII under pressure. In the second part, we identify themost probable transition pathway and validate it througha committor analysis and a histogram test46,47 on thetransition state; we then propose a quantitative valuefor the energy barrier and a mechanism of the transitionthat takes into account quantitatively the reorientationof CO2 molecules in the crystal as well as the deformation

3

of the box.

II. METHODS

To circumvent timescale limitations of standard molec-ular dynamics, enhanced sampling techniques are de-signed to accelerate the sampling of rare events. In thiswork we employ well-tempered metadynamics48 (WT-MetaD). Briefly, WTMetaD is based on the introduc-tion of a history-dependent bias potential (VG) along alow-dimensional set of collective variables (CVs)45,48–51.Such bias allows for an efficient sampling of phase space,enhancing the escape from long-lived metastable states.Significantly, this result is achieved with little a prioriknowledge of the free energy landscape, and provides anestimate of the unbiased free energy surface (FES, F(S)).For a detailed description of WTMetaD we refer the in-terested reader to Barducci et al.45,48, and Valsson etal.51, and for a brief overview of its applications in crys-tallisation studies to Giberti et al.52.

Force field

Here, we employ the rigid three-site TraPPE forcefield53,54 (Table I), with Lennard Jones potential andLorentz-Berthelot combination rules.This force field is chosen among a variety of models devel-oped for CO2

24,26,55,56, since, even if it is not tailor-madefor the high temperature and high pressure regime of in-terest, it outperforms other models in the description notonly of the liquid-vapor equilibrium at high pressures55

(up to 100 MPa), of the melting curve of dry ice (up to1 GPa) and the triple point56. Moreover, it has a betterrepresentation of the quadrupole, which is indeed rele-vant in carbon dioxide molecules and plays an importantrole in the solid phase stabilisation57.We employ two dummy atoms per molecule58 to mantainthe desired rigidity and linearity of CO2, avoiding insta-bility caused by the rigid 180◦ OCO angle. The mostrelevant limitation of this model might be the rigidity ofthe CO2 molecules41.

mC mO σC−C [nm] σO−O [nm] εC−C [kJ/mol]

12 16 0.280 0.305 0.224

εO−O [kJ/mol] qc [e] qo [e] lC−O [A] αO−C−O [◦]

0.657 0.70 -0.35 1.160 180

TABLE I. Parameters for the TraPPE force field

Simulation setup

Long-range corrections for the Van der Waals inter-actions are included through the particle mesh Edwald

(pme) method. From consistency checks on the effect ofthe cut-off value on the system volume and energy whencoupled with pme, we find that 0.7 nm is a good trade-offbetween accuracy and computational cost.

Isothermal and isobaric (NPT) simulations useBussi-Donadio-Parrinello thermostat59 and Berendsenanisotropic barostat60 for T and P control, respectively.The timestep employed is 0.5 fs.

For WTMetaD, the initial height of the Gaussians is10 kJ/mol, with width 7.81e-3 for both CVs. The bias-factor is either 100 or 200 to allow the exploration of awide portion of the phase-space. Moreover, we limit theelongation of each box side at 1.7 to 3.0 nm through theintroduction of a repulsive potential. This action pre-vents an excessive and irreversible distortion of the boxwhen the transition to melt is observed under anisotropiccontrol. We highlight that such restraints are active onlywhen the system undergoes large fluctuations in the liq-uid state. The T-P conditions investigated include ar-eas of the phase diagram where the most stable phasechanges from melt to phase I to III: at 350 K, the rangeof pressure of the present study spans from 1 to 25 GPa(1, 3, 5, 8, 12, 25 GPa). The initial configuration ofWTMetaD simulations is phase I, initially equilibratedfor 500 ps at NVT, then 5 ns NPT without pme andadditionally 5 ns NPT with pme. All simulation boxescontain supercells of 256 CO2 molecules.We perform MD and WTMetaD simulations with Gro-macs 5.2.161 and Plumed 2.262; the building of thecells and the post processing of the data employsmainly VMD63, to visualise trajectories, and MATLAB(R2015a).

Committor Analysis

As mentioned in the opening, we complement our WT-MetaD simulations with a committor analysis. Whilefor a detailed description we refer to Tuckerman47 andPeters46, we recall here some useful definitions and pro-cedures.

The committor is defined as the probabilitypB(r1, . . . , rN ) ≡ pB(r) that a trajectory initiatedfrom a configuration r1, . . . , rN ≡ r with velocitiessampled from a Maxwell-Boltzmann distribution willarrive in state B before state A47. In our study weidentify A as phase I and B as III. An important pointon the pathway connecting two basins is the transitionstate (TS, indicated with *), which is the ensemble ofconfigurations r with CV S(r)=S* that have committorpB(r) = 0.5; on free energy hypersurfaces, it correspondsto a saddle point, i.e. the highest energy state along theminimum energy path connecting two basins.To locate the saddle point, we extract 135 configurationsalong the transition pathway and for each of themwe run between 10 to 40 unbiased NPT simulationswith different initial velocities randomly generated froma Maxwell-Boltzmann distribution. Simulations are

4

stopped when they commit either basin I or III and theyare assigned an outcome value of 0 or 1, respectively.The average of the outcome values obtained from theset of trajectories generated for a given configurationprovides an estimate of the committor pIII(r) for thatconfiguration.

We have further analysed the histogram test, which,instead, studies the committor distribution, P (pB(r)),which is the probability that a configuration r withS(r)=S* has committor pB(r) = p∗. The shape of thisdistribution is a descriptor of the capability of the CVsto represent the transition mechanism: a Gaussian dis-tribution results from good CVs, while a flat or parabolicdistribution corresponds to CVs that do not describe ad-equately the transition state ensemble.To evaluate the committor probability, we consider 41configurations with CVs close to the estimated transitionstate, and for each of them, we evaluate the committor,pmIII , as previously described, and build the histogram ofP (pIII).

Collective variables

In this work, we use a CV developed by Salvalaglio etal.64–66 and employed also in Giberti et al.67. In partic-ular, every crystal structure has a unique typical localenvironment around each CO2 molecule, a fingerprintof the arrangment, and this order parameter, hereaftercalled λ, exploits this feature to effectively distinguishpolymorphs. Indeed, λ describes crystallinity, a globalproperty of the ensemble, as the sum of local contribu-tions, Γi; each Γi takes into account both the local den-sity, ρi, within a cut-off, rcut, around the i-th molecule,and the orientation, θij , respect to its neighbours (Fig-ure 2 (a)). The value of λ ranges between 0 and 1, as itexpresses the portion of molecules in the system that areordered according the geometry of a defined polymorphicstructure.A complete description of the formulation of this param-eter is reported in the Supporting Information and thecited literature.

From the characterization of the local order in poly-morphs I and III we can observe and compare peculiari-ties of the angle distribution of each phase (Figure 2 (b)and (c)), useful in the following tuning of CVs. First ofall, the arrangements of phases I and III present similar-ities, as there is overlap between the distributions of twocharacteristic angles, which are however centred in differ-ent values (in around 70.2◦ and 109.8◦ for form I, whilein around 75.6◦ and 104.4◦ for III). Moreover, phase IIIpopulates two additional characteristic angles, with val-ues smaller than 10◦ and bigger than 170◦, which mightrelate to the presence of layers. We remark also thatthe melt has a sinusoidal distribution of angles, consis-tent with a random orientation of molecules. As a finalnote, increasing temperature enhances the fluctuationsof the molecules in the crystal without modifying the

mean value of the characteristic angles; an exception tothis are the layer angles of form III that, instead, changefrom ∼1◦ to ∼8◦ and from ∼179◦ to ∼172◦ with growingtemperature.The number of neighbours in the first coordination shellshows, instead, a narrow distribution and the same valuefor the two structures, i.e. 12. Such observations lead tothe tuning two CVs, namely λI and λIII (see SupportingInformation).

Order parameter λI This CV expresses the degreeof phase I-likeness. The purpose of the tuning is tomaximize λI when the crystal structure is phase I. Toreach this aim, two characteristic angles, θk, are included,which are the ones of phase I (Table II).

Order parameter λIII Similarly, the tuning of λIIIaims at maximise the parameter in presence of phase III.However, in this case we do not talk about phase III-likeness, because as θk we select only the specific anglesthat characterize layers (Table II).

For both CVs, the cut-off rcut is set to 4.0 A, as it de-limits the first coordination shell; the width of the Gaus-sians associated to the angles, σk, instead, is in bothcases set to maximise the difference between the value ofλ in phase I and melt and it is the same for both thecharacteristic angles due to the symmetry (Table II).

θ1 [◦] θ2 [◦] σ1 = σ2 [◦] ncut [-] rcut [A]

λI 70.47 108.86 14.32 5 4

λIII 8.02 171.89 11.46 5 4

TABLE II. Tuning of the λ-order parameters. The table re-ports both the angles set, θ1 and θ2, while only one Gaussianwidth, as for symmetry reasons it is the same for both angles.The cut-off values for the number of neighbours, ncut, and thecoordination shell, rcut, are presented as well.

The phase-space evaluated on unbiased MD simula-tions suggests that the CVs are effective in the distinc-tion of separate and well-defined areas for each phase(Figure 2 (d) and (e)). Furthermore, the average orderparameters can be extracted as an ensemble average.

Temperature and pressure act on the location wherephases are projected in CV-space: on the one hand,increasing temperature decreases the values of both λswhile widening their fluctuations, consistently with thefact that the volume increases and the molecules vibratemore; on the other hand, increasing pressure leads to anincrease in the absolute value of the parameters, whilenarrowing their distribution.

5

FIG. 2. (a) Visual model of the environment around a CO2 molecule within the sphere of radius rcut. (b) - (c) Angle distributionover a range of temperature for phases I at 5 GPa (b) and III at 25 GPa (c). The green arrow points an example for melt. (d)- (e) CV-space (λI -λIII) for phase I, within the dotted blue line, and phase III, within the dotted orange line at 5 GPa overa range of T from 50 to 1000 K (d) and at 350 K over a range of P from 5 to 25 GPa (e); the arrows point the direction ofgrowing temperature or pressure.

III. RESULTS

A. Free Energy of the I-III polymorphic transition as afunction of pressure

In the following, we present the results of our study ofthe I - III polymorphic transition in CO2.Firstly, we just mention that from preliminary MD un-biased simulations phase III has a smaller volume thanphase I under all conditions investigated (∼2%, in agree-ment with experimental results); the volume predictions,however, slightly overestimate the experimental values(Figure ?? in the Supporting Information). In addition,for the same T-P settings, form I presents a lower poten-tial energy than III: the potential energy of the system isthus not a good indicator of the relative thermodynamicstability at finite temperature. We report the outcomeof MD in the Supporting Information.

Then, we discuss the results obtained from WTMetaDsimulations run with the set-up discussed before.To begin with, we observe the temporal evolution of theCVs, for the explicative case at 350 K - 5 GPa (Figure 3(a) and (b)); the other conditions investigated (Figure ??in the Supporting Information) behave in a reasonablysimilar way. In the plots in Figure 3 it is possible to

identify the system arranged in phase I as λI (a) is high(fluctuations between 0.7 and 0.9), λIII (b) is below 0.05and does not present relevant fluctuations, while the boxedges (c) have the same length. The exploration of phaseIII’s basin, instead, shows wider fluctuations in the rangeof 0.36 - 0.6 for λI (a), and 0.1 - 0.4 for λIII (b); the boxedges, moreover, fluctuates around the unbiased average.Thanks to this clarification, it is possible to spot in Fig-ure 3 that the system undergoes a significant number ofrecrossings between polymorphs I and III, in particular,four in slightly more than 5 ns at the beginning of therun. In addition, the system explores areas of the CV-space which do not represent any of these polymorphs,feature that will result more evident from the plots ofthe free energy surface (Figure 4). On the same surfacesit will be possible to notice the important role that thementioned fluctuations of the CVs have on the shape ofthe basins for the two phases.

Furthermore, by observing the output trajectoriesand data of WTMetaD, we remark two interesting be-haviours: on the one hand, CO2 molecules in the simu-lation box rearrange with a concerted motion during aphase transition; on the other hand, we find that suchtransition is anisotropic, meaning that each side of thebox is equally likely to either elongate or shorten from

6

FIG. 3. Time evolution of the CVs λI (a) and λIII (b) over150 ns, and of the box edges (c) for the first 30 ns, of WT-MetaD at 350 K - 5 GPa.

I to III (Figure 3 (c)). In particular, this latter ob-servation is important since by biasing as CVs orderparameters that account for the spacial orientation ofmolecules, we obtain the consequential deformation ofthe supercell, without considering the box volume oredges as CVs, as instead done in previous MetaD workson polymorphism5,13–16,18–20.

Next, we present the free energy surfaces (FES) recon-structed by WTMetaD. In such FESs, the free energy isexpressed as a function of the CVs: G(λI , λIII) (Figure 4(a) and (f) to (i)). Before proceeding with the discussion,we underline that the FES at 25 GPa is not reported, asno recrossing is sampled from phase III; in addition, theresults at 1 GPa are taken into account only qualitatively,since under such conditions phase III is so unstable thatspontaneously evolves to I in standard MD and it is thusnot possible to locate its basin.Some considerations can be drawn from the study of theFESs. First of all, the location of the minima on the FESfor phases I and III is accurately close to the predictionin Figure 2(d)-(e). Moreover, phase III has a much widerbasin than phase I and it develops mainly along λIII ,while phase I’s mainly along λI , as underlined for thetemporal evolution of the CVs (Figure 3). As mentionedbefore, the system explores a wide area of CV-space and,

in particular, the presence of black boxes in Figure 4 high-lights the presence of defected phase I structures, whichwe shall analyse in detail later on. Relevant structuralarrangements are reported in Figure 4 (b) to (e).

In order to compare the results of WTMetaD with theexperimental phase diagram, we study quantitatively therelative stability between polymorphs.Keeping in mind that the free energy is a function ofthe probability distribution of the CVs, it is possible toevaluate ∆GI−III as (1):

∆GI−III = GI −GIII = −β−1 ln

(ppIppIII

)(1)

Where ppI is the probability of phase I, ppIII of phaseIII, and β is 1/kT. The probability of each phase is com-puted as the integral of the distribution within the basinit occupies on the CV-space:

ppI =

∫

I

p(λ) dλ =

∫∫

λI ,λIII∈Ip(λI , λIII) dλI dλIII

(2)

ppIII =

∫

III

p(λ) dλ =

∫∫

λI ,λIII∈IIIp(λI , λIII) dλI dλIII

(3)The integration domains are identified by coloured boxeson the FES in Figure 4 (a) and (f) to (i). In Figure 5 (a)we report relevant ∆G values over the range of pressureconsidered. The relative stabilities in Figure 5 (a) to-gether with the FESs in Figure 4 allow to draw some con-siderations on the phase diagram. We observe that whilethe boundary of the solid - melt transition is in goodagreement with experiments, the I - III transition pres-sure appears underestimated. From the ∆GI−III patternshown in green in Figure 5 (a), the transition pressure at350 K can be estimated as around 4.5 GPa. Despiteunderestimating the experimental value, the transitionpressure agrees with literature results obtained treatingCO2 as a rigid molecule38–40. We also recall that com-monly experimental works rather than a single value re-port a transition pressure interval (see Introduction), towhich our estimation is closer.Nevertheless, it is possible to notice that WTMetaDsimulations are able to represent the overall trend ob-served in the phase diagram: increasing pressures in-crease the stability of phase III, while at decreasing valuesof pressure phase I is more stable, ultimately reaching theboundary with melt.Since the behaviour of solid carbon dioxide is so welldescribed, it is possible to consider a translation of thephase diagram.

A further step in the analysis of the I - III relativestability is the breakdown of the free energy in its inter-nal energy, mechanical work and entropy contributions.With this aim, we firstly evaluate the difference in in-ternal energy, ∆U and mechanical work, P∆V , betweenphase I and phase III from the ensemble averages com-puted from the unbiased MD simulations; the entropy is

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FIG. 4. FESs at 350 K under a range of pressure: 1 GPa (f), 3 GPa (g), 5 GPa (a), 8 GPa (h), and 12 GPa (i). The colour barspaces for all surfaces from 0 to 1400 kJ/mol. Blue boxes locate the basin of phase I, orange phase III, melt is within a greenbox, while black rectangles identify phase I with defects. In addition, (a) reports also the minimum free energy transition path(red). The structures reported are III in (b), I in (c) and two examples of packing faults in (d) and (e); the letters are an aidto compare the packing. Only one plane is displayed as the most explicative of defects; however, while one of the not shownplanes is almost perfect, the other has the correct motif, but the layers are not perfectly aligned.

thus obtained from the macroscopic definition of Gibbsfree energy:

∆G = ∆U + P∆V − T∆S (4)

From the results in Figure 5 (b) it possible to noticesome major features. First of all, the internal energy,∆U , stabilizes form I, while P∆V is significant in thestabilisation of phase III; in both cases their contribu-tion becomes more relevant with growing pressures. Theentropic term, instead, tends to favour form I, a partfrom pressure of 12 GPa.

Defected phases

As mentioned in the analysis of the CVs (Figure 3)and of the FESs (Figure 4), at pressure equal and above

3 GPa, the system evolves to new, not a priori knownphases, which we recognise being defected structures I(Figure 4 (d) and (e)). Indeed, such phases are simi-lar to phase I, being almost cubic and having compara-ble arrangement; however, they display packing faults,planar defects that break the orientation motif recog-nisable in perfect phase I. The perfect arrangement, infact, presents the repetition of rows of CO2 that alternatethe orientation respect to the Cartesian axis in a sort ofABABABAB sequence (Figure 4 (c)), while the defectedphases replicate two or more lines with the same “charac-ter”, AA or BB (Figure 4 (d) and (e)). It is particularlyremarkable the capability of WTMetaD to predict theproduction of defected structures, as, despite this phe-nomenon can takes place in experiments, it is “underrep-resented in the current literature”69, due to its difficultcharacterization both experimentally and through mod-elling.

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FIG. 5. Relative stability between different phases (a) and breakdown of the free energy (b) at 350 K and increasing pressurefrom 3 to 12 GPa. In (a) green squares represent ∆G between phase I and III, red between perfect phase I and defectedstructures I, while blue bewteen a comprehensive phase I, including configurations with and without defects, and phase III.Positive values of ∆G mean that phase III (green, blue) and defected I (red) are more stable. The error bars are obtainedfrom a weighted averaged on simulation time, similarly to Berteotti et al.68. In (b) the focus in on the contributions to therelative stability between phases I and III: the ∆G obtained by WTMetaD is again plotted in green, yellow represents theinternal energy difference of the two phases from MD, red their difference in mechanical work from MD, while blue the entropydifference obtained from the definition of Gibbs free energy (Eq 4); we report the entropic contribution as -T∆S, so that forall the terms considered negative values stabilize phase I and positive phase III. In both graphs, dashed lines are an aid to theeye to visualise the trend.

In addition, we observe that the stability of the defectedphases increases at higher pressures, becoming ultimatelyeven more stable than phase I (∆G in red in Figure 5 (a)).This behaviour may be due to a more difficult expansionof volume from phase III to I under higher pressure, andthus defected phases with a smaller volume form.

To complete the analysis of these phases, we run un-biased simulations under the same T-P conditions as therelated WTMetaD, and with the defected arrangementof interest as initial configuration. The results show thatthese forms do not spontaneously undergo any transition:the creation and correction of defects is thus an activatedevent.

B. Committor analysis

In the first part of this work, we have shown that our λ-order parameters are effective CVs in sampling the tran-sition between polymorphs I and III, evaluating their rel-ative stability, and exploring the phase space. In the fol-lowing, we focus instead on the mechanism of the titletransition. In particular, such analysis allows to eval-uate the goodness of the CVs in the representation ofthe process, and to estimate quantitatively the transitionpathway and the energy barrier to overcome.

First of all, we characterize the minimum free energypath (MFEP) that connects the free energy minima cor-

responding to phase I and III. The MFEP provides arepresentation in CV-space of the most probable set ofintermediate states involved in the transition. Further-more, the free energy profile along this path yields anestimate of the free energy barrier associated to the poly-morphic transition. As initial estimate of the MFEP wepropose an approximation obtained as the combinationof the projection of FES along the CVs, more precisely,of basin I along λI and of basin III along λIII , due tothe observe typical L-shaped FES; further details aboutthese approximations are provided in the Supporting In-formation. We then employ such path as educated guessfor an optimisation routine that enables to obtain theactual MFEP from a series of trial moves, whose accep-tance is based on the free energy value. The algorithmis robust and the path converges to the same route fromdifferent and less educated initial guesses.

In Figure 4 (a) we report the MFEP on the FES at 350K - 5 GPa, while in Figure 6(a) we compare transitionpathways evaluated at different pressures. Interestingly,pressure only slightly affects the typical L-shape of thetransition pathway, with the major difference being thelocation of the minima. Moreover, at this level of detail,the energy barrier to overcome from phase I to III ap-pears similar at all pressures investigated (Figure 6(b)).We also highlight that the MFEP converges much earlierthan the simulation, and no alternative routes connect-ing polymorphs I - III arise (Figure S7 in the Supporting

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FIG. 6. (a) Transition pathway in the space of CVs at 350 K over the range of pressures investigated. (b) Projection of thefree energy along the curvilinear path coordinate at 350 K over a range of pressures. The progression along the MFEP spacesbetween 0 in phase III and 1 in phase I. The minimum of phase I’s basin is the free energy reference. In both graphs blue dotsrepresent phase I, while orange III.

Information).The next step towards a quantitative characterization

of the transition mechanism is the validation of MFEPthrough a committor analysis46,47, with histogram teston the apparent transition state. We discuss hereafterthe explicative case of 350 K - 5 GPa.

To begin with, we locate the transition state by evalu-ating the committor of 135 configurations extracted alongthe MFEP. Interestingly, configurations with committordifferent from zero and one are not evenly distributedalong the transition pathway, but grouped in a narrowarea around the saddle point, estimated in λ∗I = 0.65,λ∗III = 0.034; as a result, the cumulative distributionalong the path resembles a very steep sigma shape (Fig-ure 7(a)). The behaviour shown in Figure 7(a) suggeststhat the order parameters alone might not be enoughto account for the transition mechanism. The validationproceeds with a histogram test on the saddle point. Weevaluate the committor of 41 configurations with λ(r)around λ∗ and represent the results on a histogram (Fig-ure 7 (c)). Such histogram shows three peaks, sign thatour CVs alone are not effective reaction coordinates andother parameters need to be included in the mechanismdescription.

In order to identify the additional parameters to takeinto account, we deepen our analysis and further inves-tigate the dependence of pIII(r) on properties such aspotential energy, volume and box dimensions (Figure S6in the Supporting Information). The results suggest thatthe deformation of the lattice plays a role in the repre-sentation of I - III transition mechanism. We define thisdeformation through the simulation box anisotropy, i.e.the ratio between the longest and the shortest sides of

the cell; its value spans from 1 in cubic phase I to 1.35in orthorhombic phase III. As a result, we note that onlyconfigurations r along the pathway with anisotropy of thebox between 1.14 and 1.145 have committor non identicalto 0 or 1, and, in particular the TS is uniquely locatedin λ∗I = 0.65, λ∗III = 0.034, anisotropy∗ = 1.1421, whisecharacteristic orientations are presented in Figure 7 (f).We thus repeat the histogram test on 19 configurationswith CVs and anisotropy close to the TS and the outcomeshows, as expected, a Gaussian shape (Figure 7 (d)): asa result, to effectively describe the mechanism of the I -III transition of solid CO2 all three parameters, namelyλI , λIII and anisotropy, have to be taken into account.

We thus evaluate70 the free energy as a function ofthe three parameters of interest: G(λI , λIII , anisotropy).On such FES we identify the 3D MFEP that connectsphase I to phase III (Figure 7 (b)): its projection onthe λI − λIII plane reasonably overlaps with the MFEPpreviously evaluated; moreover the anisotropy of the boxmonotonically increases from I to III, and vice versa.

Summing up the analysis carried on in this secondpart of the work, the transition from cubic phase I toorthorhombic phase III can be thus described as the se-quence of the following actions:

• The CO2 molecules firstly tend to distort the typi-cal phase I lattice and, as a consequence, the valueof λI decreases, with no relevant increase of λIII(horizontal branch of the L-shaped pathway); atthe same time the box starts deforming, elongatingone side and reducing the others, thus increasingits anisotropy.

• Then, when the deformation of the cell reaches the

10

FIG. 7. Committor analysis results, with the transition mechanism described only by the CVs (a,c), or by the CVs and thebox anisotropy (b,d-f). In (a) coloured dots represent configurations, r, extracted along the MFEP (black) on the CV-space;their colour is based on the committor of each configuration, shading from black for pIII(r) = 0, to red for pIII(r) = 1. Thesame colour code applies to the insert in (a), which reports the committor as a function of the progression along the MFEP;in the same graph, the sigmoid dashed line is an aid to the eye to read the trend. The histogram test on configurations rso that λ(r)=λ∗ of the TS shows three peaks (c). The inclusion of the box anisotropy to the λ-order parameters to describethe mechanism requires a 3D representation and thus in (b) we report the free energy as a function of these three parametersthrough a colour plot and the transition path through red dots; the results of the histogram test (d) on configurations with thesame λI , λIII , and anisotropy of the TS confirms that this set of parameters is effective and complete. In (e) the free energyis plotted as a function of the progression along the 3D path highlighted in (b), with reference in phase III. An orange dotlocates phase III (at progression zero along the path), a blue dot phase I (at progression 1), and a red dot the transition state;representations of the structures of phase I, phase III and the TS are within blue, orange and red rectangles, respectively. Thesame colour code is employed for the angle distributions for phase I, phase III and the TS presented in (f).

anysotropy threshold value of the transition state,the system completes the rearrangement to phaseIII; indeed, the molecules start organizing into par-allel layers and the volume decreases. From this

point the transformation proceeds on the verticalbranch of the L-shaped pathway, with increasingλIII for relative small variations of λI .

The motion of the molecules in the crystal during the

11

transition is concerted.From the investigation of the 3D MFEP we obtain also

quantitative information about the height of the barrierfor the polymorphic transformation: for a system com-posed by 256 CO2 molecules at 350 K - 5 GPa the tran-sition state is located at about 202 kJ/mol (Figure 7(e)),with reference zero in phase III, i.e. the absolute mini-mum of the FES.

IV. CONCLUSIONS

In this work we presented an investigation of the I -III polymorphic transition in carbon dioxide under pres-sure. Our approach combines molecular dynamics, well-tempered metadynamics and committor analysis to pro-vide a broad insight into this phenomenon.

Firstly, we performed WTMetaD simulations at 350K over a range of pressure (1 - 25 GPa) with two or-der parameters as CVs. These parameters, λI and λIII ,are built on the local order around each CO2 moleculeand account for the reorientation of the molecules in thecrystal. This feature allows to clearly distinguish in theλI -λIII CV space configurations that belong to phase Ior phase III and to clearly resolve amorphous configura-tions. Moreover, metadynamics exploration with theseCVs allows to sample the formation of packing faults inphase I. Interestingly, we observe the deformation of thecell, a global rearrangement of the configuration, takingplace as a consequence of enhanced sampling along λIand λIII , which account for local order. This also per-mitted to notice that, in a I to III transition, all sides ofthe box have the same probability to elongate or shorten.

From the FESs resulting from WTMetaD, we evalu-ated the free energy difference between polymorphs; weobserved that the predicted trend of the I - III relativestability over pressure is in agreement with the carbondioxide phase diagram: increasing pressures move fromthe melt-phase I boundary, to the region of stability ofphase I, to the one phase III. We estimate the tran-sition pressure at ∼4.5 GPa, in agreement with previ-ous literature works that considered carbon dioxide as arigid molecule38–40. Furthermore, our model suggest thatthe stability of the defected configurations increases withpressure. While at low pressure undefected form I is morestable than the ensemble of its defected counterparts, athigh pressure the latter appears to dominate.

Alongside a description the I-III transition thermody-namics, we assess the I-III polymorphic transition mech-anism for the representative case at 350 K - 5 GPa. Tothis aim we identify the MFEP connecting phase I tophase III in CV space and we validated the pathway car-rying out committor analysis and histogram test on anensemble of configurations corresponding to the saddlepoint in CV space. From this analysis it emerges thatto quantitatively identify the transition mechanism weneed to consider the anisotropic deformation of the CO2

supercell alongside order parameters accounting for the

local arrangement of CO2 molecules. This analysis al-lowed to identify a reliable approximation of the tran-sition pathway, and hence to quantify the free energybarrier associated with the transition.

Our work shows that, by combining opportunely de-signed order parameters with state-of-the-art enhancedsampling methods and committor analysis, we can pro-vide an in-depth characterisation of both thermodynam-ics and transition mechanisms of polymorphic transfor-mations at finite temperature.

SUPPLEMENTARY MATERIAL

See supplementary material for further details on thecollective variables, and additional results on MFEP,committor analysis, and unbiased simulations.

ACKNOWLEDGEMENTS

The authors acknowledge EPSRC (Engineering andPhysical Sciences Research Council) for PhD scholarship,and UCL Legion High Performance Computing Facilityfor access to Legion@UCL and associated support ser-vices, in the completion of this work.

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Supporting Information: CO2 packing polymorphism under pressure:mechanism and thermodynamics of the I-III polymorphic transitiona)

Ilaria Gimondi1 and Matteo Salvalaglio1, b)

Department of Chemical Engineering, University College London.

(Dated: 16 April 2018)

SUPPORTING INFO

1. Collective Variables

In this work, we run WTMetaD employing as CVs aset of two λ order parameters, namely λI and λIII . Suchparameters were first developed and used by Salvalaglioet al.1–3 and Giberti et al.4 in the study of nucleation andmelting of urea. We refer to these works while presentinghere further details on the formulation of the parameters.

As mentioned in the main manuscript, λ representsthe degree of crystallinity of the system as a sum of localcontributions, Γi (Eq (1)).

λ =1

N

N∑

i=1

Γi (1)

Dividing by the total number of molecules in the system,N, ensures that λ, similarly to a molecular fraction, ex-presses the portion of particles organized alike a definedcrystal structure, ranging from 0 to 1.As said, each Γi considers the local order around the i -th CO2 molecule in terms of density, ρi, and orientationwith respect to neighbours, θij . First of all, the localdensity, ρi, is based on the coordination number, ni: ifthe number of neighbours of i is bigger than the cut-offncut, the molecule is crystal like (Eq (2)).

ρi =1

1 + e−b(ni−ncut)(2)

where b tunes the slope of the switching function and niis a function of the distance rij between the i-th and thej-th molecules and a well-defined cut-off rcut (Eq (3) andEq (4)):

ni =N∑

j=1j 6=i

fij (3)

fij =1

1 + ea(rcut−ri)(4)

where a tunes the slope.A second feature considered is the orientation between

a)Footnote to title of article.b)Electronic mail: m.salvalaglio@ucl.ac.uk

FIG. S1. Preliminary run at 600 K - 5 GPa considering λIas the only CV. The results show that this parameter aloneis unable to distinguish the basins of phase III and melt.

neighbouring molecules, i.e. the angle between i and thej -th molecules within rcut from i, θij . This angle θijin a crystal fluctuates around the characteristic orienta-tions θk according to a Gaussian distribution (with widthσk); the term Θij accounts for this behaviour (Eq (5)).To consider only orientation between neighbours, Θij ismultiplied by fij (Eq (4)).

Θij =

kmax∑

k=1

e− (θij−θk)

2

2σ2k (5)

Overall, the local crystallinity Γi is expressed by Eq (6)and tends to 1 when the molecule is in the descriptedsolid phase:

Γi =ρini

N∑

j=1

fijΘij (6)

The use of tunable switching functions allows to havecontinuous and differentiable CVs.

In the present work we employ a set of two CVs,namely λI and λIII . The necessity of two Cvs is due tothe fact that in preliminary test runs with only λI , thebasin of phase III overlaps with the one of the melt whenit is formed close to the solid-liquid border (Figure S1).

2. Unbiased MD

We hereafter report the results in terms of volume (Fig-ure S2) and potential energy (Figure S3) for unbiased MD

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FIG. S2. Trend of the specific volume in cm3/mol over tem-perature at different pressures (5, 12, 25 GPa) for phase I(squared markers) and phase III (diamond markers). Thedark blue dot represents the value obtained for melt at 800 K- 5 GPa from the spontaneous evolution of phase I.

performed for phases I and III over a range of pressure(5 - 25 GPa) and temperature (50 - 700 K). The set up isthe same described in the main text, a part from phase I,for which we employed a cut-off radius of 1.0 nm. Fromthese simulations we obtain the equilibrated initial con-figurations for WTMetaD as well as an insight into theproperties of the system.

The general trend for volume and potential energy is thesame for both polymorphs: volume and potential energygrow with temperature, while at growing pressures, thepotential energy increases and the volume decreases. Fo-cusing on the volume (Figure S2), orthorhombic phase IIIhas a smaller volume than cubic I, such difference beingaround 2 %, in agreement with the literature. MOre-over, the comparison of the length of the unit cell edgesof phase I with experimental results5,6 (Figure S4) showsthat the set up with the TraPPE force field tends to over-estimate its value, with a smaller deviation the smallerthe pressure. Our values at 295 K are interpolated fromthe reasonably linear trend shown by volume in Fig-ure S2.The width of the distribution of potential energy, instead,increases with temperature and such enlargement of thefluctuations leads to an interesting overlap of the poten-tial energy distribution of the two polymorphs. How-ever, increasing pressure drastically reduces the extentof the overlay. Moreover, the potential energy is higherfor phase III in all the conditions simulated.

In conclusion of the presentation of MD results, wehighlight that no transition between different phases issampled, unless in conditions of high overheating (phaseI to melt at 800 K - 5 GPa, in agreement with the predic-tions and observations of Perez-Sanchez et al.7) or highunder-pressurizing (phase III to I at 350 K - 1 GPa).This is a further confirmation that polymorphic transi-

tions are rare events, which require enhanced samplingtechniques to be thoroughly sampled. In addition, weremark that, by observing the potential energy distribu-tions of phases I and III, such property by itself is not areliable indicator of the thermodynamic stability.

3. WTMetaD results

In Figure S5 (a-b), we report the fluctuations of thevalue of volume of the box and potential energy dur-ing the first polymorphic transition at 350 K - 5 GPa,compared with the averages obtained in unbiased MD(dashed lines); as the results are in good agreement, WT-MetaD gives an accurate description of the system.Then, Figure S5 (c-h) plot the temporal evolution of λIand λIII at 350 K and 3, 8, 12 GPa. Similar observationsas for the case reported in the main text (Figure ??) canbe drawn, in particular for the high number of conver-sions and the “anisotropic” transitions.

4. Approximation of MFEP

As educated guess of the MFEP we proposed acombination of two approximations of the transitionpathway in CV space. Such approximations arenamely G(λI , λIII)|λI=const, the locus of the minima inG(λI ,λIII) at constant λI , and G(λI , λIII)|λIII=const,the locus of the minima in G(λI ,λIII) at constantλIII . The necessity to combine these representationsis due to the characteristic L-shaped FES, which causesG(λI , λIII)|λI=const to be a better approximation of theMFEP within basin I, while G(λI , λIII)|λIII=const withinbasin III (Figure S6 (a)).We then performed an optimisation routine that, start-ing from an initial guess, attempts replacements of posi-tions along the path with adjacent ones along specifieddirection: the trial move is accepted if the free energyis smaller than the initial guess and the coordinates aredifferent from the previous and following along the path;this procedure is iterated until convergence of the MFEP.The final path obtained is the same from different initialsets (Figure S6(b)). We notice that the MFEP is verysimilar to the combined path in the minima basins, whileit describes more accurately the portion in between them(Figure S6(d)-(f)).The transition pathway converges to the MFEP well be-fore the convergence of the simulation (Figure S6(c)) andno alternative routes emerge.

5. Committor Analysis

In order to better understand the area on the CV-spacearound the saddle point, we investigate the dependenceof pIII on properties other than the local order, in par-ticular on potential energy, volume and anisotropy of the

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cell (Figure S7); the anisotropy is defined here as the ra-tio between the longest and shortest box edge. Whilethe committor of a configuration does not seem to showany correlation with its potential energy or volume, theclear and steep sigma-shape that pIII shows as a func-tion of the box anisotropy (Figure S7 (c)) suggests thenecessity to include this property in the characterisationof the transition state.

I. REFERENCES

1Matteo Salvalaglio, Thomas Vetter, Federico Giberti, Marco Maz-zotti, and Michele Parrinello. Uncovering Molecular Details ofUrea Crystal Growth in the Presence of Additives. Journal of theAmerican Chemical Society, 134(41):17221–17233, oct 2012.

2Matteo Salvalaglio, Thomas Vetter, Marco Mazzotti, and MicheleParrinello. Controlling and Predicting Crystal Shapes: The

Case of Urea. Angewandte Chemie International Edition,52(50):13369–13372, dec 2013.

3Matteo Salvalaglio, Claudio Perego, Federico Giberti, Marco Maz-zotti, and Michele Parrinello. Molecular-dynamics simulations ofurea nucleation from aqueous solution. Proceedings of the Na-tional Academy of Sciences, 112(1):E6–E14, jan 2015.

4Federico Giberti, Matteo Salvalaglio, Marco Mazzotti, andMichele Parrinello. Insight into the nucleation of urea crystalsfrom the melt. Chemical Engineering Science, 121:51–59, jan2015.

5Bart Olinger. The compression of solid CO2 at 296 K to 10 GPa.The Journal of Chemical Physics, 77(12):6255, 1982.

6Lin-gun Liu. Compression and phase behavior of solid CO2 to halfa megabar. Earth and Planetary Science Letters, 71(1):104–110,nov 1984.

7G. Perez-Sanchez, D. Gonzalez-Salgado, M. M. Pineiro, andC. Vega. Fluid-solid equilibrium of carbon dioxide as obtainedfrom computer simulations of several popular potential models:The role of the quadrupole. The Journal of Chemical Physics,138(8):084506, 2013.

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FIG. S3. Potential energy of phase I (blue) and phase III (orange) in kJ/mol, expressed through its probability distributionover a range of temperature and pressure.

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FIG. S4. Comparison of the unit cell side (a) and volume (b) of phase I with experimental results from Olinger5 and Liu6.

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FIG. S5. Consistency check of the box volume (a) and the potential energy (b) obtain from WTMetaD with the unbiasedvalues (dashed lines) of phases I (blue) and III (orange) for the explicative case at 350 K - 5 GPa; time evolution of the CVsat 350 K and 3 GPa (c-d), 8 GPa (e-f) and 12 GPa (g-h).

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FIG. S6. (a) Comparison between transition pathways G(λI , λIII)|λI=const (red line) and G(λI , λIII)|λIII=const (black line)and their combination (dashed green line) at 350 K - 5 GPa. (d)-(f) Initial guess pathway (black continuous line) and finalMFEP (grey stars and line) obtained from our optimisation routine, starting from the combined path, G(λI , λIII)|λI=const(red line) and G(λI , λIII)|λIII=const, respectively. The comparison of the final results is reported in (b), which shows that thepaths converge to the same actual MFEP. The time evolution of the MFEP is reported in (c), where the path colour changesfrom black to red with increasing simulation time.

FIG. S7. Representation of the committor probability pIII over potential energy (a), volume (b) and cell anisotropy (c)