Dynamical and structural correlation in supercooled liquids: A molecular dynamics investigation of...

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 1 1 JULY 2003

Dynamical and structural correlation in supercooled liquids:A molecular dynamics investigation of m-toluidine

Riccardo Chelli, Gianni Cardini, Piero Procacci, Roberto Righini, and Salvatore Califanoa)

Dipartimento di Chimica, Universita` di Firenze, Via della Lastruccia 3, 50019 Sesto Fiorentino, Italyand European Laboratory for Non-linear Spectroscopy (LENS), Via Nello Carrara 1,50019 Sesto Fiorentino, Italy

~Received 14 January 2003; accepted 1 April 2003!

Structural and dynamical properties of liquid and supercooled liquid m-toluidine are studied bymolecular dynamics simulations. Approaching the liquid–glass transition, dynamical hetero-geneities, a characteristic common to all supercooled glass formers, are observed. We prove theoccurrence of strict correlation between these heterogeneities and the potential energy landscape ofthe system, expressed in terms of molecule–molecule interactions. A slowing down of theself-diffusive motion of the molecular centers of mass is observed for particular arrangements ofpairs of H-bonded molecules. Previous studies on model systems provided evidence of thecorrelation between dynamical heterogeneities and potential energy landscape, described in terms ofthe inherent structure of the system. While in this last case the structure is viewed as a collectiveproperty of the system, in m-toluidine short-range interactions are sufficient to explain thedynamical behavior in a satisfactory way. This result agrees with the view, supported also byexperimental observations, that m-toluidine can be considered as an ensemble of H-bondedsubsystems weakly interacting among them. ©2003 American Institute of Physics.@DOI: 10.1063/1.1576754#

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I. INTRODUCTION

The nature of supercooled liquids and their tendencyform glassy states represent still unsolved problems of cdensed phase physics. For these liquids self-diffusion perwell below the melting temperature, up to the so-called gltransition temperature (Tg), where transport propertiechange by many orders of magnitude. Several theoraimed to explain the liquid–glass transition, have beproposed1–3 to incorporate the dynamical aspect of the liuid. In the last decade the mode coupling theory~MCT! be-came very popular for the description of the dynamicalpects of the liquid–glass transition. In its simplest versio4

MCT is devised for a fluid of shapeless particles, for whithe only dynamical variables are the positions of the cenof mass. The earliest version, the idealized MCT, predicdynamical transition between two regimes at a critical teperatureTc : in the first regime (T.Tc), the time correlationfunction of any density function decays to zero in a fintime, while in the second one (T,Tc), all these functionsremain finite at infinite time, this limit being called thenonergodicity parameter. In particular, whenTc is ap-proached by cooling, transport properties are characterby a non-Arrhenius temperature dependence, and the stural relaxation~a-relaxation! has a marked nonexponentibehavior, characterized by a stretched exponential evoluof the related correlation functions.5 On the basis of a number of experiments6–9 and simulations,10–15 it is now gener-ally accepted that the nonexponential relaxation is due topresence of dynamic heterogeneity, i.e., to a broad distr

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tion of particle mobilities throughout the liquid. This behaior is well accounted for by MCT aboveTc , where thestretched exponential behavior of the correlation functionpredicted to be scaled in a master curve.4 However, when theMCT predictions are compared with experimental data,proximate agreements can be obtained only ifTc is largerthanTg ~see Ref. 4!. Many authors argue that the breakdowof the MCT predictions might arise from the increasing importance of hopping events~not accounted for in the simplesversion of MCT! in the vicinity of the glass transition. Thepresence of hopping events and their importance in demining the time relaxation of dynamical quantities has bestudied for various glass-forming model systems.15–17 Inthese studies the analysis of the dynamics of model systwas performed using the concept of potential enelandscape18 ~PEL! and the correlated idea ofinherentstructure.19,20Although the PEL treatment includes all detaof the interaction potential of the system, its applicationcomplex molecular liquids to obtain local structure informtion is actually difficult due to the large computational coof the method. In principle, interaction potential details~mo-lecular shape, for example! could be also included in theMCT formalism.21–23 On the other hand, the reverse wanamely the extraction of the local structure data fromMCT predictions, cannot be followed. In this framewormolecular dynamics~MD! simulation can be considered aone of the most useful tools to study the local structure ofliquids.

Previous investigations that aimed to understand thenamical heterogeneity in supercooled liquids were maidevoted to study model systems,12–15 and application tosimple molecular liquids has started to appear24 only very

© 2003 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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358 J. Chem. Phys., Vol. 119, No. 1, 1 July 2003 Chelli et al.

recently. Studies on model systems~in many cases LennardJones fluids! are very attractive because their simplicity alows long simulation runs and makes the analysis easierthe other hand, the high simplicity of the model is an ostacle to the understanding of the microscopic structumechanisms leading to the dynamical behavior of thesystems.

For a more realistic approach, we presented in a recpaper25 a MD simulation of a truly molecular liquidm-toluidine, one of the most studied glass-forming molecusystems. We suggested that the appearance of dynamicaerogeneity in liquid m-toluidine is strictly related to the ocurrence of extensive structural rearrangements, involvpairs of H-bonded molecules. From an analysis of the fenergy surface of these dimers, we concluded that their mprobable molecular arrangement goes from T-shaped~withperpendicular displaced molecular planes! at high tempera-ture, to stacked~with parallel displaced molecular plane!below 260 K. In the present paper we investigate the naof both the structural and the dynamical heterogeneityterms of the radial-angular pair distribution functions andthe time-dependent mean-square displacement of thelecular centers of mass. In addition we include in the discsion a new MD simulation at 210 K, very close to the glatransition temperature of 190 K obtained in Ref. 25. Frothe analysis of the atomic trajectories, we show that itpossible to interpret the observed dynamical heterogeneiterms of well-defined subsystems, consisting of stronglyteracting molecules. This treatment is consistent withview, proposed by other authors,15 that a system with a largenumber of particles can be decomposed into weakly intering subsystems with little correlation among each other.the specific case of m-toluidine, the minimal subsystemswhich the dynamical properties of the system stronglypend, have a dominant contribution from pairs of H-bondmolecules.

The paper is organized as follows: in Sec. II we briedescribe the potential model used for the simulations andessential technical details. In Sec. III the results are repoand discussed. Conclusions are given in Sec. IV.

II. POTENTIAL MODEL AND TECHNICAL DETAILS

The MD simulations were performed using an all-atomodel. The molecules were assumed completely flexiblemeans of aCHARMM-like potential.26 The intermolecular po-tential is modeled by point charges on the atoms andLennard-Jones interactions. A complete description ofintra- and intermolecular potential is given in Ref. 25. Tsimulated sample consists of 256 molecules in a cubic bDetails of the equilibration procedure were also givpreviously.25 For the additional 210 K simulation run, thsame equilibration and coordinate-storage protocols usedthe 220 K simulation were adopted.25 Most of the results wepresent to discuss the different behavior of m-toluidinefrom and close to the liquid–glass transition refer to the 3and 210 K MD simulations, respectively.


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III. RESULTS AND DISCUSSION

Although m-toluidine could not be properly considereas one of the strongest H-bond forming compounds like wter or glycerol, several of their chemical physics propertcan be explained only considering its attitude to formbonds. For example, there are experimental and theoreinvestigations that indicate the H-bond induced clusteringthe origin of the appearance of an unusual prepeak instatic structure factor of m-toluidine.25,27–29 In this respect,we found25 evidence of a relevant structural change, closeTg , by analyzing the reciprocal arrangement of pairsH-bonded molecules. Strong indication of the role playedH-bonding was also revealed by the high percentageH-bonded molecules,25 which ranges from 46% at 340 K to84% at 210 K. Because of the importance of H-bondm-toluidine in general, and for the quantities calculatedthe present work in particular, we recall the H-bonding denition used in the previous paper25 and also adopted here. Ausual, in the absence of a rigorous definition, a geometrcriterion is used.30–33 A H-bond between two m-toluidinemolecules is formed if the N H distance is less than 2.5 Åcorresponding to the distance of the first minimum of the Nradial distribution function~see Fig. 1 of Ref. 25!, and if theN¯HN angle is larger than 120 degrees. We assume abond to be formed/destroyed if the two criteria mentionare fulfilled/not fulfilled for at least 240 consecutive fs. Thidea of introducing a lifetime criterion in the H-bond defintion is about 2 decades old,34,35 and more recently it wassupported by Sciortino and Fornili in a study of the timdependence of pair interactions in liquid water.36

As discussed in the Introduction, the goal of this papeto show that the structural heterogeneity of the supercooliquid phase is correlated to the dynamical heterogeneSuch a correlation is conveniently studied using two quaties easily derived from MD simulations: the mean-squdisplacement of the molecular centers of mass as a funcof time, which provides information about the dynamicsthe system, and the radial–angular pair distribution functwhich provides information about its structure. This laquantity is defined as follows:

g~R,u!5^( i 51

N ( j Þ iN d~R2Ri j !d~u2u i j !&

2p sinu R2 DR Du N2 V21 , ~1!

where the angular brackets indicate a time average, the son i and j run over all theN molecules of the simulationsample,d(x) is the Dirac function,Ri j and u i j are the dis-tance between the centers of mass and the angle formethe normals to the aromatic rings of thei and j molecules,respectively,V is the simulation box volume, andDR andDuare the resolutions in distance and angle ofg(R,u), respec-tively. For all the functions depending onR andu reported inthe following, we usedDR50.2 Å andDu52.5 deg.

Of interest for the present discussion is essentiallycontributionghb(R,u) to g(R,u) of the pairs of H-bondedmolecules. For simplicity, only theghb(R,u) functions at210 and 300 K are shown in Fig. 1. At high temperaturedominant broad peak, labeled A in the figure, is observedR;6 Å and u;90 deg. This peak corresponds to dime


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359J. Chem. Phys., Vol. 119, No. 1, 1 July 2003 Structure and dynamics of supercooled m-toluidine

FIG. 1. ~Color! Radial–angular pairdistribution function, ghb(R,u) forH-bonded molecules at 210 and 30K. The reference values for the chromatic scale are reported on the righside of each picture.

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structures in which the axes of the aromatic rings are perpdicular to each other. As the temperature is lowered the pbecomes sharper and sharper, due to the increased H-association and to the localization of preferred structures210 K, new peaks, labeled B, C, and D in Fig. 1, appeTheir intensity is even larger than that of A. Structures cresponding to the peaks C and D have instead a nearlyallel orientation of the two rings. Representative structuremolecular pairs belonging to the A, B, C, and D regionsFig. 1 are shown in Fig. 2. From the figure it is easily sethat structures corresponding to peak B, C, and D defrom A by tilting in part or completely one ring with respeto the other. For the structures corresponding to the A, B,D peaks the molecule–molecule interactions are almostclusively limited to H-bonding. In the C structure the ringring and ring–methyl interactions give instead an import

FIG. 2. Representative structures of the A, B, C, and D (R,u) regions ofFig. 1.


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contribution. We notice that the strong peak C at 210 K ispresent at 220 K, as can be seen from the free energy lscape at 220 K reported in Ref. 25. In addition, the pecorresponding to the D region becomes at 210 K musharper than at 220 K. The increase with decreasing tempture of the probability of parallel displaced configuratiowas also observed in Ref. 25, where it was attributed tstructural transition. While the A, B, and D dimers are chacterized by large exposed surfaces, the C structure is mcompact, and similar to that observed for the minimum eergy dimer.25 This confirms the hypothesis25 that, by lower-ing the temperature, the H-bonded dimeric structuresfrom entropy-favored to energy-favored configurations. Wshall show below that the presence of H-bonded pair strtures, like B, C, and D, is decisive for the appearancedynamical heterogeneity in supercooled liquid m-toluidine

As previously discussed, the dynamical properties ofsystem have been analyzed by calculating the mean-sqdisplacement of the molecular centers of mass as a funcof time37

D~ t !51

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^uRi~ t !2Ri~0!u2&. ~2!

In Eq. ~2!, Ri(t) is the vector position of the center of maof the i th molecule,N is the number of molecules in thsample, and the angular brackets indicate an average ontime origins. TheD(t) function for the MD simulations at340, 300, 260, 220, and 210 K is shown in Fig. 3. The sloof D(t) is related to the translational self-diffusion coefcient of the center of mass. Our MD model yields at 21220, 260, 300, and 340 K the values 0.009, 0.027, 0.83.873,38 and 9.941~units of 1026 cm2 s21), respectively.From 340 to 210 K the self-diffusion coefficients vary bmore than two orders of magnitude.

Since it is not possible to characterize all kinds of dnamical heterogeneity on the basis of collective propertsuch asD(t), some space partitioning is needed. In the


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erature there is not a general consensus on what kinparticle aggregates are involved in the cooperative motiat the origin of the dynamical heterogeneity. While seveauthors individuate the origin in compact regions of tspace,39,40 other found it in string-like structures whoslength increases with decreasing temperature.13 In principle,one should first use some computational method to chaterize the dynamical heterogeneities and then individuateregions of space~subsystems! involved in different dynami-cal processes. We proceed instead in the reverse waychosea priori and somehow arbitrarily, the putative regio~subsystems! of space and then we show that different susystems have different dynamical behavior. For this purpothe simulation box was divided into 27 equal cubic box~subsystems! and all molecules found in each subsystemthe first step of the simulation run were considered to fothat subsystem during the full simulation time. The indvidual subsystems contain from 9 to 11 molecules each.then possible to associate a time-dependent mean-squarplacement to each subsystem by using Eq.~2!, whereN rep-resents now the number of molecules of the consideredsystem. In Figs. 4~a! and ~b!, the time-dependent meansquare displacements~same as Fig. 3!, normalized by theD(t) value at 0.6 ns, are reported for the 300 and 210 K Msimulations, respectively. In the same figures, the vertbars indicate the spread of the time-dependent mean-sqdisplacements obtained for the various subsystems.curves are spread on very wide ranges. Not surprisingly,relative spread increases with decreasing temperature. Alowest temperature~210 K! one can identify three subsystems where the translational diffusion is almost frozwhile the mobility is still relatively large in many other subsystems at the same temperature. For the subsystems chterized by slow motion, the mean-square displacementthe centers of mass as a function of time are shown in4~c! along with those relative to the three subsystems wthe highest mean-square displacement~fast motion sub-systems!.

In the Introduction we posed the question whetherobserved heterogeneity in the dynamical behavior of diffent subsystems reflects also a different H-bond connectof the members of the subsystems themselves. The answthis question is given in Fig. 5, where theghb(R,u) pair

FIG. 3. Mean-square displacementsD(t) of the molecular centers of masas a function of time, at 210, 220, 260, 300, and 340 K. For the sakclarity, theD(t) functions at 210 and 220 K are reported in the inset.


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distribution function for the molecules belonging to the thrslow and to the three fast motion subsystems is reported.three slow motion subsystems, i.e., subsystems 4, 5, andFig. 4~c!, show sharp and intense peaks corresponding toregions D, B, and C ofghb(R,u) ~see Fig. 1!, respectively. Inparticular, the peaks observed for subsystems 4 and 6 cospond to the most intense D and C peaks. No similar cospondence is observed for the fast motion subsystems@sub-

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FIG. 4. ~a! Mean-square displacementD(t) as a function of time~same asFig. 3! normalized by theD(0.6) value for the 300 K MD simulation. Thevertical bars indicate the spread of the time-dependent mean-squareplacement curves for the 27 subsystems of the sample.~b! Same as~a! butrelated to the 210 K MD simulation.~c! Time-dependent mean-square diplacements for the six~fast and slow motion! subsystems at 210 K. Curve1, 2, and 3 correspond to the subsystems with the largest self-diffucoefficient, while curves 4, 5, and 6 correspond to the subsystems withlowest self-diffusion.



FIG. 5. ~Color! Radial–angular pair distribution function,ghb(R,u), for the molecules belonging to the six~fast and slow motion! subsystems at 210 K. Thereference values for the chromatic scale are reported on the right side of each picture.

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systems 1, 2, and 3 in Fig. 4~c!#, which show only broad anddisperse peaks on the (R,u) plane.

To generalize the observations made for the slowfast motion subsystems, i.e., to extract a quantity thatcludes all the molecules of the sample in the statistics,calculated a function describing the microscopic mobilitythe system in the form

P~R,u!5^( i 51

N ( j Þ iN ~di1dj !d~R2Ri j !d~u2u i j !&

2( i 51N ( j Þ i

N d~R2Ri j !d~u2u i j !. ~3!

The symbols in Eq.~3! have the usual meaning, anddi is anapproximation to the translational self-diffusion coefficieof the i th molecule, calculated according to the Einsteformula37

di51

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wheretmax is the total simulation time. TheP(R,u) functionis shown in Fig. 6 for the 210 and 300 K MD simulation


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Large values of theP(R,u) function, for one given (R,u)point, mean fast translational motion for the molecules whcontribute to that point. At 300 K a noisy unstructured shapof P(R,u) is observed, indicating that no preferentistructural–dynamical correlation is present at that tempeture. On the contrary, clear evidence of dynamical heteroneities is found in the low temperature plot. The freezingthe translational motions corresponding to the C and Dgions of Fig. 1 is well evident in this case.

In the past, interesting studies based on MD simulatioand on the inherent structure approach were carried ouSciortinoet al.41,42 to elucidate the relation existing betweemobility and structure in water. The authors found that,lowering the density, water exhibits an increase of the focoordinated molecules at the expenses of the mocoordinated ones. In addition, they observed a lower moity for the four-coordinated than for the five-coordinatemolecules. This behavior, not usual for liquids, was in geeral attributed to the lower binding energy of the fiv


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FIG. 6. ~Color! P(R,u) function at210 and 300 K. The reference valuefor the chromatic scale~in Å2 ns21!are reported on the right side of eacpicture. The statistically relevan(R,u) points, deduced byghb(R,u) ofFig. 1, are those included in the regiodelimited by blank lines. The (R,u)points without statistics are reported idark blue for the data at 300 K, whilethey are reported in brown at 210 K, tevidence low self-diffusion in the re-gion R,4 Å, u.140 deg.

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coordinated molecules, but interestingly the authors fouthat five-coordinated molecules have a faster diffusion ewhen binding energy is strong. Apparently, this result donot agree with our findings, as we find that strong pair bining energy between m-toluidine molecules can be relateslow diffusion of them. In our opinion, such a disagreemcould be attributed to the fact that Sciortino and co-workdid not include the contribution to the~de!stabilization en-ergy of the molecule–cage subsystems given by the intetion between molecules belonging to the first coordinatshell. In fact, we believe that five-coordinated H-bondclusters could have an intrinsic energetic destabilizationto steric factors, and that this destabilization could explthe observed faster diffusion of the five-coordinated wamolecules. However, to validate this hypothesis, additiocomputational investigations would be necessary.

IV. CONCLUSIONS

The idea that at low enough temperature the dynamica ~supercooled! liquid is essentially determined by the struture of the potential energy landscape~the inherent struc-tures! goes back more than 30 years to the workGoldstein.18 This approach has been recently confirmed bMD computer simulation of a model glass-former;19 alongthe same line, the explanation of the dynamical heterogeities in supercooled liquids in terms of the properties ofpotential energy landscape has been demonstrated15 for amodel system consisting of Lennard-Jones particles. Welieve that our MD simulation provides clear evidence of sua strong correlation between dynamical and structural prerties in a real system like supercooled m-toluidine. Tmain structural features of the deep valleys appearing infree energy landscape of this molecular liquid in the supcooled regime have been identified from the calculation,shown to be determined by the formation H-bonded molelar pairs with a well-defined conformation. The formationenergetically stable H-bonded molecular pairs causes a s


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decrease of the molecular mobility in those deep minimwhere the concerted motion of the H-bonded pairs becomore and more hindered. This is the source of the dynamheterogeneities that are responsible for the strongly nonponential structural relaxation observed in m-toluidine ovebroad temperature range.43

ACKNOWLEDGMENTS

We thank Renato Torre for interesting discussions. Twork was supported by the European Union under ContraERB-FMGE-CT950017 and HPRI-CT1999-00111, andthe Italian Ministero dell’Istruzione, dell’Universita` e dellaRicerca~MIUR!.

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