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Chemical Physics Letters 633 (2015) 41–46

Contents lists available at ScienceDirect

Chemical Physics Letters

jou rn al hom epage: www.elsev ier .com/ locate /cp le t t

olecular dynamics simulation of the glass transition in,4′-N,N′-dicarbazolylbiphenyl

lexey Odinokova,∗, Alexandra Freidzona,b, Alexander Bagaturyantsa,b

Photochemistry Center, RAS, Novatorov Str. 7a, 119421 Moscow, RussiaMoscow Engineering Physics Institute, Kashirskoe hwy 31, 115409 Moscow, Russia

r t i c l e i n f o

rticle history:eceived 19 March 2015

n final form 2 May 2015

a b s t r a c t

Viscoelastic properties of the molecular liquid consisting of 4,4′-N,N′-dicarbazolylbiphenyl (CBP)molecules near the glass transition temperature are investigated by molecular dynamics simulations.

vailable online 20 May 2015The relaxation dynamics is analyzed by considering each molecule as a point-like oriented particle.The dependence of the calculated properties on the coarse-grain parameter used in the calcula-tion of orientation correlation is analyzed. The divergence of ˛-relaxation times is described by theVogel–Fulcher–Tammann law and the mode coupling theory. The basic concepts of the glass transitiontheory are applied to a real amorphous organic semiconductor.

© 2015 Published by Elsevier B.V.

. Introduction

Development of a complete theory of disordered state andlass transition presents a challenging task in condensed matterhysics. A number of different approaches to explaining vitrifica-ion phenomena have been proposed during last decades [1–3].ometimes they lead to analytical findings that can be related toxperiment by the fit of parameters. Microscopic properties ofnterest can also be extracted from the numerical simulation ofhe model of amorphous material. The typical objects for computerimulations are spin-glasses and structural glasses consisting ofennard–Jones particles. Example calculations can be found in Refs.4,5], but a great amount of other publications is also available.hese test systems are simple enough to allow for high compu-ational efficiency and clear interpretation of the obtained data.owever, rapid growth of computer power makes calculations

nvolving hundreds of thousands of particles routine nowadays.his motivates researchers to apply theoretical insights to the realystems occurring in chemical physics and material science [6].or the sake of computational efficiency multiatomic molecules

re often treated in simplified manner, albeit with account of theiron-spherical shape [7–9]. It seems interesting to us to combineheoretical view of glass transition with all-atom representation of

∗ Corresponding author.E-mail address: ale.odinokov@gmail.com (A. Odinokov).

ttp://dx.doi.org/10.1016/j.cplett.2015.05.013009-2614/© 2015 Published by Elsevier B.V.

the liquid within the framework of the ‘general’ force field. Thisapproach offers a unified way to investigate chemically specificproperties of supercooled molecular liquids and amorphous solids.

In the present work we use molecular dynamics simula-tions to describe a process of glass transition in 4,4′-N,N′-dicarbazolylbiphenyl (CBP). Amorphous materials consisting ofCBP and its derivatives are typical hole-conducting organic semi-conductors [10]. They are widely used in the construction ofSMOLEDs (Small Molecule Organic Light-Emitting Diodes). Themain goal of our study was to simulate divergence of relaxationtimes during cooling and to extract essential parameters inherentto some popular theoretical models.

2. Theory

2.1. Order parameter for glass transition

Below we briefly define the key quantities describing glass tran-sition in the liquid of spherical particles of the same type. Consider avolume V with N particles inside. The density operator of the systemis

�(�r, t) =N∑

ı(�r − �ri(t)

), (1)

i=1

where �ri(t) is a position of the ith particle at the moment t. The orderparameter that governs a process of glass transition is defined as[11]:

4 Physics Letters 633 (2015) 41–46

Q

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�

Tidc˛pt

wd

w

we

Q

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2 A. Odinokov et al. / Chemical

(t) =∫

d�r1d�r2�(�r1, 0)�(�r2, t)ı(�r1 − �r2) =N∑

i=1

N∑j=1

ı(�ri(0) − �rj(t)).

(2)

The function Q(t) describes a degree of correlation between twoonfigurations separated by a time interval t, i.e. the number of par-icles that either remained at the same position, or were replacedy another particle. The generalized susceptibility can be definedhrough the magnitude of the fluctuations of the order parameter11]:

4(t) = ˇV

N2

[⟨Q 2(t)

⟩−

⟨Q (t)

⟩2]

. (3)

his function has a complex shape but eventually it reaches its max-mum at some point, after which decays monotonically. This decayesignates complete relaxation of the density autocorrelation andan be considered as a microscopic counterpart of the macroscopic-relaxation. Further, we will denote by �˛ the point where �4(t)ossesses its maximum value. This quantity will be used as a mainool to describe divergent time scale of the structural relaxation.

In order to compute Q(t), one has to replace Dirac delta in (2)ith some finite counterpart. In practice, the ‘overlap’ function isefined:

(∣∣�r1 − �r2

∣∣) ={

1,∣∣�r1 − �r2

∣∣ ≤ a

0,∣∣�r1 − �r2

∣∣ ≥ a,(4)

here a is a coarse-grain parameter. In terms of function w thexpressions (2) and (3) can be rewritten as [11]:

(t) =N∑

i=1

N∑j=1

w[�ri(0) − �rj(t)], (5)

4(t) = ˇV

N2[〈Q 2(t)〉 − 〈Q (t)〉2]. (6)

For every specific value of a all thermal vibrations with the mag-itude less than a is disregarded, so the fast decay of correlation onhort times is not taken into account. This parameter also affects thealue of Q(t) and �4(t) for larger time and space scales, but all phys-cally meaningful results originate from their time dependence andherefore remain the same in some interval of values of a.

.2. The case of multiatomic molecules

If the liquid is composed of the multiatomic molecules thenheoretical framework from previous section has to be reconside-ed. Strictly speaking, the formulas from Section 2.1, when appliedo the positions of individual atoms, can describe vitrificationhenomena if the difference in atomic types is taken into account.ut such an approach, along with computational inefficiency, mixesotion of the molecule as a whole with intramolecular motions of

toms, which can hinder analysis.With this in mind, we decided on another approach. Confor-

ational freedom of CBP molecule is limited to the rotationsround torsional angles ϕ1 and ϕ2, with overall shape of theolecule remaining almost the same. From now on we disregard

ll intramolecular motions and assume that material consists ofhe non-spherical elongated particles. Position �ri of every parti-le corresponds to the center of mass (COM) of the molecule andrientation �ni is defined as the normalized vector connecting two

Figure 1. Molecular structure (top) and relaxed PES scans for two torsional anglesof CBP (bottom).

nitrogen atoms. To use the results of the previous section let usintroduce the ‘overlap’ function for orientation vectors as follows:

u(

∠(�n1, �n2

))=

{1, ∠

(�n1, �n2)

≤ b

0, ∠(�n1, �n2

)≥ b,

(7)

with a new coarse-grain parameter b. The definition of the orderparameter in Eq. (5) changes:

Q (t) =N∑

i=1

N∑j=1

w[�ri(0) − �rj(t)

]u(

∠(�ni(0), �nj(t)

)). (8)

This approach is based on the assumption that the mutualrearrangement of molecules is the most sluggish and large-scalemotion in the system and therefore is responsible for all thephenomena near glass transition.

3. Computational methodology

3.1. MD simulation details

All interactions in the system were parametrized using OPLSforce filed [12]. The charges were calculated through ESP fittingprocedure at the MP2/6-31G* level of theory. The majority of theparameters for bonded interactions were borrowed from indoleand benzene species. Parameters for torsional angles ϕ1 and ϕ2 (seeFigure 1) were fitted to the potential energy surface (PES) scans, cal-culated at the MP2/6-31G* level of theory. All quantum-chemicalcalculations were performed with the aid of Firefly QC package [13],which is partially based on the GAMESS (US) [14] source code.

The GROMACS [15] package was used for all molecular dynamicssimulations. The time step of 2 fs was used, length of all hydrogenbonds were constrained to constant value via LINCS [16] algorithm.

Velocity rescaling thermostat [17] with time constant 0.3 ps wasapplied to the system. When the calculation methodology requiredconstant pressure simulation, the Berendsen barostat [18] withtime constant 1 ps and compressibility = 4.5 × 105 bar−1 was also

A. Odinokov et al. / Chemical Physics Letters 633 (2015) 41–46 43

ubate

3

pospmadnticrfseltItasF

4

4

lt2ccI

Figure 3. Expansion of the simulation box with a rise of temperature. The huge

Figure 2. Simulation box with 200 CBP molecules.

sed. Van-der-Waals interactions were smoothly switched to zeroetween 1.0 nm and 1.2 nm. Cut-off length for electrostatic inter-ctions was also 1.2 nm, while the long-range part was treatedhrough the PME algorithm [19]. Dispersion correction [20] fornergy and pressure was applied throughout all calculations.

.2. Simulation workflow

In our calculations we generated the structure of amor-hous material with the so-called ‘melt-quench’ method. First webtained a model of the liquid phase at high temperature, then theystem was slowly cooled down by lowering the reference tem-erature. The structure of dense liquid was created as follows. CBPolecules were successively inserted in the cubic simulation box

t random place with random orientation. If the overlap of Van-er-Waals spheres was detected, the result of the last step waseglected and a new try was made. This process was repeated untilhe box contained the desired number of molecules. This algorithms incorporated into the GROMACS program package. The successan be achieved only for densities much less (∼3 times) than theeal one. That’s why at the next stage a special MD run was per-ormed at the high temperature (∼900 K). During this run the boxize was decreased monotonically until the system approached thexperimental density of crystal. After that the equilibration andong production run were carried out in the NPT ensemble at theemperature much higher than the glass transition point (∼600 K).ndividual snapshots from the obtained trajectory, separated by aime interval longer than the time of ˛-relaxation, were regardeds totally uncorrelated and served as starting points for all furtherimulations. The general view of the simulation box is shown inigure 2.

. Results

.1. Crossing glass transition temperature

To observe glass transition directly in computer simulation aong MD simulation has been performed. During this simulationhe box with 200 CBP molecules inside was cooled from 500 K to

00 K. The total length of the trajectory was 20 ns, it produced theooling rate of 1 × 1010 K s−1. The process of glass transition can beharacterized by more or less abrupt change in thermal expansivity.t can really be seen from a plot of cell volume versus temperature

amount of points gathered throughout simulation is denoted as a continuum densityvia contour lines.

(Figure 3) that the volume has linear asymptotic form on both lim-its (liquid and solid glass). The point of glass transition Tg can beestimated from the crossing point of two corresponding lines. Itsvalue Tg = 369 ± 5K is 34 K higher than the experimental one (335 K[10]).

4.2. Calculation of dynamical susceptibility

To compute order parameter for liquid with Eq. (8), one has todefine coarse-grain parameters a and b. Their values have to belarge enough to cut small thermal vibrations but still noticeablyless than typical inter-particle distance. The value of a = 0.3 timesinter-particle distance seems to be good choice for atomic liquids[21,22]. For molecular liquid we can regard the position of the firstpeak of the radial distribution function between the centres of massof the molecules as a measure of the inter-particle distance. Judgingby this criterion we obtain a = 0.4 nm. The choice of b is much moreambiguous. We could not find any counterpart in the literature. Wecan assume by analogy that the upper boundary for this parameteris 90◦, as the conformations become totally uncorrelated. Also it isreasonably to suppose that the choice of b within some interval doesnot affect physically meaningful quantities. In all our calculationswe considered b = 15◦. Some numerical tests and other reasoningbehind this decision are given in Section 4.3.

The set of MD simulations was performed for the isotropic boxwith 200 CBP molecules. Each run corresponded to the specific tem-perature above Tg. Altogether, the temperature range from 450 Kto 750 K was covered. The time dependence of the susceptibility�4(t) is given in the Figure 4. As the temperature goes down andapproaches the point of glass transition, the position of the maxi-mum of �4(t) and its maximum value increases. The same pictureis common for the monoatomic glass-forming liquids [23,11].

4.3. Analysis of the singularity in relaxation times

The long-time behavior of �4(t) is determined by the rateof ˛-relaxation processes in the medium. After some specificpoint, susceptibility curve decays monotonically to its asymp-totic value. We denote the ˛-relaxation time as �˛ and identifyit with the position of the maximum of �4(t). This quantity

diverges at low temperatures, being effectively infinite for solidglass. Several numerical recipes describing this singularity behav-ior has been proposed based on the different theoretical models

44 A. Odinokov et al. / Chemical Physics Letters 633 (2015) 41–46

[V

�

wo[

�

wplttfi

Table 1The values of the parameters from Eqs. (9) and (10) obtained via fitting.

b TVFT D TMCT �(K) (K) (K)

10◦ 310 4.85 402 3.88

Figure 4. The overlap susceptibility for different temperatures.

23,24,3]. We used two well-known approaches. First is theogel–Fulcher–Tammann (VFT) law:

˛ ∝ exp(

DTVFT

T − TVFT

), (9)

here D is a constant and TVFT is a critical temperature. The sec-nd approach is the prediction of the mode-coupling theory (MCT)25,26]:

˛ ∝ (T − TMCT )−� , (10)

here � is a positive exponential factor and TMCT is a critical tem-erature. The results are represented in Figure 5 in the form that

inearizes dependencies (9) and (10). The response of the results to

he change of parameter b was also investigated. Apart from b = 15◦

wo other values of b was used: 10◦ and 20◦. All data obtained bytting is given in Table 1. It should be noted that these calculations

Figure 5. Dependence of the ˛-relaxation time on temperature.

15◦ 319 3.87 405 3.3520◦ 321 3.81 404 3.45

were made with the same MD trajectories, so the differences inTable 1 were originated only from the different value of b used.

It can be seen from Table 1 and Figure 5 that results for b = 15◦

and b = 20◦ almost coincide, while results for b = 10◦ slightly dif-fers. This fact has a close connection with a problem of the intervalof legitimate values of b. Autocorrelation function of molecularorientation decays near-exponentially with time, but at the verybeginning it suffers a fast drop (see Figure 6). This drop can beattributed to the motions of the molecule inside its cavity (ˇ-relaxation) and to parasitic fluctuations of the orientation vectordue to the thermal vibrations of nitrogen atoms. Another possi-ble factor is the limitations of the model of rigid elongated particlewe used. In reality, molecules can bend and change inner geome-try. This has an unpredictable impact on the effective value of �˛.While boundary values of 15◦ and 20◦ cut off problematic motions,the value of 10◦ is situated close to the amplitude of the initialdrop and the results become corrupted. We can discard the firstraw of Table 1 and after averaging propose: TVFT = 320 K, D = 3.84,TMCT = 405 K and � = 3.4.

5. Discussion

5.1. What can be achieved using the present approach?

Incorporation of orientational effects into the density correla-tion (8) allowed for the successful computation of the dynamicalsusceptibility. The form of its peaks near glass transition temper-ature resembles the behavior of its counterpart for the system ofspherical particles. The obtained values of the parameters Tg, TVFT

and TMCT have a right order at the temperature axis. Although Tg

overestimates the experiment, the discrepancy is expected becauseof the very fast cooling rate used in simulations. Moreover, empiri-cal force-filed was developed for simulations at normal conditions

and there is not guarantee of its applicability in the wide rangeof temperatures. Therefore, the model of rigid oriented particleswe used can be recommended for description of glass transitionin molecular liquids. It should be noted, however, that this model

Figure 6. Short-time behaviour of the rotational autocorrelation function of molec-ular orientation.

Physics Letters 633 (2015) 41–46 45

ishcd

a(dwmqafhdstrc

5s

siop[siaoamitltml

asmumspp

pwnpict[fcist

Figure 7. Angell plots for different glass-formers. Calculated data for CBP are shownwith open circles. Full lines correspond to VFT curves plotted with parametersborrowed from the following sources: SiO2, B2O3, PbO·SiO2, PbO·B2O3 – Ref. [44],

A. Odinokov et al. / Chemical

mpose severe restrictions on the type of the molecules under con-ideration. In many cases molecules of organic semiconductorsave rather complex shape and undergo major conformationalhanges. In such a situation some intramolecular degrees of free-om must be included in the model.

Using molecular dynamics simulations we obtained reason-bly long trajectories for the system consisted of 200 molecules12400 atoms). Total sample size allows for computation of theensity correlation and corresponding dynamical susceptibility,hich are the functions of time and contain the structural infor-ation only in averaged form. The theory of glass transition offers

uantities that depend on larger number of space and time vari-bles, such as scattering function or four-point density correlationunction. Knowledge of their dependence on temperature couldelp us to explore a lot of challenging issues concerning disor-ered matter, e.g. spatially heterogeneous dynamics [27]. But theampling was too poor and the calculation of higher-order func-ions was impossible. To reach this goal much more computingesources need to be spend, which is troublesome even for modernomputers.

.2. Is the structure of supercooled liquid an adequate model forolid glass?

The melt-quench method used in our work represents the mosttraightforward and methodologically simple approach to model-ng the structure of disordered matter. The examples of applicationsf this method are numerous. It was used to simulate amor-hous silicium [28,29], silica-gel [30] and silica-gel-based glasses31], chalcogenide glasses [32,33], germanium alloys [34,35], boronemiconductors [36] and even quasi-crystals [37]. The high qual-ty of the result is usually gained only if the structures of solidnd liquid phases are alike in some sense. The common featuref the examples given above is that the atoms in such substancesre connected with a net of directed covalent or ionic bonds. Theutual arrangement of atoms reveals a short-ranged order, which

s observed as a set of relatively sharp peaks of radial distribu-ion function (RDF). If coordination numbers of atoms or typicalocal bonding patterns are different for liquid and solid states,hen quenching fails to reproduce the structure of rigid amorphous

aterial. This artifact is widely known and is reflected in partiallyiquid-like solid state [32].

In light of this, melt-quench approach can be considereds a promising method for simulation of amorphous organicemiconductors. These materials consist of individual organicolecules, which are connected with relatively weak intermolec-

lar interactions. Due to the large size and complex shape eacholecule has a lot of irregular contacts with neighbors. A noticeable

hort-ranged order is not supposed to exist both for liquid and solidhases. Molecules show structureless mutual arrangement, whichrovides a possibility to model solid glass by simple quenching.

On the other hand, glass structure is mainly governed by theroduction method. Organic semiconductors exist as thin filmshich are not produced by quenching. The list of possible tech-ologies includes vacuum deposition [38], spin-coating [39] andrinting [40]. It is difficult to predict what structural features can be

ntroduced by every specific methodology [41]. In many cases opti-al properties of the material become anisotropic, which indicateshat the molecules are partially aligned in the preferred direction42]. Moreover, many organic semiconductors are bad glass-ormers and recrystallize after melting and cooling at the moderate

ooling rates. Therefore, CBP glass investigated in the current studys actually a limiting case of the quasiequilibrium supercooledtates of the liquid. It can be considered as a simplified model ofhe isotropic bulk disordered solid, which is used in the absence

t-PG (tripropylene glycol) – Ref. [45], m-toluidine – Ref. [46], TPD (N,N′-bis(3-methylphenyl)-N,N′-diphenylbenzidine) — Ref. [47].

of detailed information on the structure of the real amorphousmaterial.

5.3. A comparison of CBP with other glass-formers

Although glass transition point for different liquids may varyover a wide range of temperatures, there is a convenient wayto compare their viscoelastic properties upon cooling. A plot ofthe logarithm of viscosity (or relaxation time) as a function ofTg/T (Angell plot [43]) is often used to divide liquids into twogroups: ‘strong’ and ‘fragile’ glass-formers. ‘Strong’ glass-formershave almost linear (Arrhenius) behavior, while ‘fragile’ display sub-stantial nonlinearity. The difference can be expressed via parameterD in the VFT law (9). The larger the value of D, the more ‘strong’ isglass.

It was noted that unknown coefficient in Eq (9) often weaklydepended on the material [44]. Following Sethna [48], let us assumethat it equals 10−14 s. After accepting this hypothesis, we are ableto draw Angell plots for various glass-forming liquids along withour calculated results in the same figure (see Figure 7). Computersimulations provide very limited degree of proximity to the glasstransition point, so our results present only tail of the total curveas compared with experiment. However, both numerical value ofD (3.84, see Table 1) and the position of the calculated pointson the Angell plot show that CBP falls into the category of ‘frag-ile’ glass-formers. This behavior is common for molecular liquids.Some examples are given in Figure 7. Similar properties have beenobserved experimentally [47] for another small molecule organicsemiconductor at temperatures much closer to the glass transition(see dotted line in Figure 7).

6. Conclusion

We performed molecular dynamics simulation of the liquidCBP near the glass transition. In order to analyze relaxationdynamics, the obtained trajectories were remapped to representa model system of oriented point-like particles correspondingto individual molecules. Using this approach, we computed thedynamical susceptibility and the corresponding relaxation times.

The divergent relaxation times were described in terms of theVogel–Fulcher–Tammann law and the mode coupling theory. Anextension of the present model to other materials consisting ofmolecules of arbitrary shape requires a further development. An

4 Physi

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6 A. Odinokov et al. / Chemical

lternative approach implies that the density operator is basedn the positions of all atoms. This approach does not impose anyrtificial constraints on the system but has a high demand foromputational resources. An amount of operations required forhe computation of the density correlation grows as the square ofhe total number of particles. In our calculations, the most time-onsuming stage was MD simulation, while all the further analysisook only a few percent of the total time. If we used an atom-basedensity, the calculations would become impractically long. This

ustifies the application of the molecule-based density to organicemiconductors and other molecular liquids. Further developmentn this area offers prospects for the use of efficient computational

ethods in the studies of amorphous materials.

cknowledgements

This work was supported by the Russian Science Foundation,ontract no. 14-43-00052. The computational facilities were pro-ided by the Joint Supercomputer Center of the Russian Academyf Sciences.

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