Molecular dynamics simulation of a molecular glass at intermediate times

Physica A 201 (1993) 150-156 North-Holland

SDI: 037%4371(93)E0247-C

Molecular dynamics simulation of a molecular glass at intermediate times

G&an Wahnstriim” and Laurent J. Lewisb ‘Department of Applied Physics, Chalmers University of Technology and University of Giiteborg, S-412 96 Giiteborg, Sweden

bDkpartement de physique et Groupe de Recherche en Physique et Technologie des Couches Minces, VniversitC de Mont&al, C.P. 6128, Succursale A, Montreal, QuCbec H3C 3J7, Canada

The results of a detailed molecular dynamics investigation of relaxation in the van der Waals system o-terphenyl are presented. Our calculations give clear evidence for the existence of a fast “relaxation process” on the picosecond time scale. This confirms the neutron-scattering observation of an anomalous decrease of the Debye-Wailer factor. The observed relaxation process results from strongly anharmonic motion of molecules trapped in cages formed by their neighbours.

The dynamics of glass formation is currently an area of intense investigation

(see, for instance, various articles in [l]). One of the most studied simple glass

forming substances is o-terphenyl (1,2_diphenylbenzene), a nonpolar organic

liquid. It consists of well-defined molecular units (three connected benzene

rings) and the intermolecular interaction is of the short-range van der Waals type. It shows little tendency to crystallization, with a melting temperature T,

of 329 K. The system can easily be supercooled; the corresponding structural

relaxation is slow and (as in other glass forming systems) referred to as

primary, or (Y, relaxation. The calorimetric glass transition temperature is Tg = 243 K.

Inelastic neutron scattering measurements have been performed on o-terphenyl, demonstrating the existence of a secondary relaxation mode, on the picosecond time scale [2-41 (i.e., much faster than the conventional secondary relaxation process seen in dielectric measurements, the /3 process [5,6]). We will denote this fast process pr, following ref. [7]. The Pr process gives rise to an anomalous decrease of the Debye-Waller factor at temperatures close to 290 K [2,3], substantially higher than the calorimetric T,. Depolarized light scattering experiments also reveal the existence of a fast process, with an essentially temperature independent correlation time (of about 3 ps) in the

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G. Wahnstriim, L.J. Lewis I Molecular dynamics simulation of molecular glass 151

temperature range 250-450 K [8,9]. Below Tg also nuclear magnetic resonance measurements reveal a fast mode on the picosecond time scale [lo].

Motivated by these experimental findings, and in an attempt to better understand the atomic scale processes that take place during pr relaxation, we have performed an extensive series of molecular dynamics (MD) simulations. In order to make the model for o-terphenyl simple, thus tractable, we represent it by a three-site molecule, each site playing the role of a whole benzene ring. Molecules interact additively pairwise with one another, with interactions between two sites on different molecules of the Lennard-Jones form, u(r) = 4~[((+lr)‘~ - (~/r)~]. Each molecule is assumed to be rigid, with a bond length g and a bond angle of 75 degrees. Explicit values for E and u were determined by adjusting to the experimental values for the molar volume and the diffusion coefficient at T = 400 K and the following values were obtained: E/kg = 600 K and u = 4.83 8, [ll]. The number of molecules in the simulation

is 324 (972 sites). A quantity of central importance in the understanding of glassy dynamics is

the intermediate scattering function, F(q, t). In conventional neutron scattering experiments, its time Fourier transform, the dynamical structure factor S(q, w), is measured directly. In the case of o-terphenyl, the scattering is essentially incoherent, due to the high proton content, and we will therefore concentrate on the incoherent (or self) part, F”(q, t).

Using our MD data we have determined the intermediate scattering function for the site-site correlations. (Site-site correlations in our model correspond to neutrons scattering off the centres of the different benzene rings.) Typical results are shown in fig. 1 for the incoherent part F”(q, t). When temperature is lowered, the decay in time slows down dramatically and pronounced stretching is observed. Around T = 270 K, F”(q, t) does not decay to zero for times less than 10e9 s, the time window in our simulations. The system is “structurally arrested” on this time scale. Proper equilibration cannot be achieved for these temperatures. In a previous simulation [12] we have shown by performing three independent set of calculations that this is not a serious issue.

In the neutron scattering studies it was found that the slow decay, which gives rise to a quasielastic peak in the spectrum, is well approximated by a Kohlrausch-Williams-Watts (KWW) behaviour. In the time domain the KWW law corresponds to a stretched exponential, F”(q, t) = a exp[-(tlr)P], and in ref. [2] a, p, and 7 were determined at different temperatures and for different wave vectors by fitting to the experimental spectra. We have performed a similar analysis [ll]; the corresponding stretched exponentials are shown as dashed lines in fig. 1.

The temperature dependence of the amplitude parameter a = a( q, T) has attracted considerable interest because of the claim by mode coupling theory

152 G. Wahnstriim, L.J. Lewis I Molecular dynamics simulation of molecular glass

time (ps)

Fig. 1.

l~~IIIIIIII.IIIIIIIIIII,I 90 130 170 210 250 290 330

temperature T (K)

Fig. 2.

Fig. 1. Self-intermediate scattering function F”(q, t) versus time for wavevector q = 1.94 8-l at various temperatures; the dashed lines are fits to the stretched exponential a exp [-(t/v)‘].

Fig. 2. The amplitude of the fitted stretched exponentials a = a(q, T), the “non-ergodicity parameter”, versus temperature for the three wavevectors indicated. The dashed lines correspond to the usual linear dependence in the low-temperature harmonic regime, and full lines to square root prediction of mode-coupling theory near T, = 280 K.

that it is the signature of an underlying instability in the system [13]. The results of our MD evaluation of this quantity are shown in fig. 2. At low temperatures, a(q, T) is well described by the corresponding expression for a harmonic system, viz. In a = -Wq2T (where W is a constant independent of both q and T), as indicated by dashed lines in fig. 2. However, for T > 250 K, the system exhibits substantial departure from linearity, i.e., becomes strongly anharmonic.

Very similar behavior is in fact observed in the neutron scattering measurements [2] and we conclude that our simple model for o-terphenyl contains the anomalous decrease of the Debye-Waller factor (or more correctly the Lamb- Mdssbauer factor), seen experimentally.

Mode coupling theory predicts a temperature dependence of the form

a(q, T) = a,(q) + h(q)[T, - WTc11’2 for T < T,, with T, a system dependent

“critical temperature”. A good fit to our data (indicated by full lines in fig. 2) is obtained with T, = 280 K. Our calculations, therefore, are consistent with the predictions of mode coupling theory for the existence of a “dynamical glass transition temperature”, T,, at a temperature somewhat above the calorimetric

T,. The extra loss of the elastic intensity, associated with the anomalous

G. WahnstrBm, L.J. Lewis I Molecular dynamics simulation of molecular glass 153

decrease of the Debye-Waller factor, reappears as a flat background in neutron scattering spectra [2]. In fig. 3 we show, as the full line, the dynamical susceptibility x”(q, w) = wS’(q, w) at T = 275 K in a log-log plot, determined from our MD data. The dashed line shows the combined contribution from the microscopic motion (defined by processes faster than 3 ps) and the (Y relaxation (the stretched exponential). The difference between the dashed and full lines, the broad featureless structure in the frequency range lo-100 GHz (0.04- 0.4 meV), therefore corresponds to the excess relaxation associated with the anomalous decrease of the Debye-Waller factor.

We use the words “excess relaxation” for the fact that the scattering function shows an additional reduction compared with the harmonic behaviour. We have not been able to clarify if this excess relaxation should be regarded as a vibrational type of motion or more a structural relaxation process [14]. The distinction between these two kinds of processes is not at all evident.

Depolarized light scattering experiments on o-terphenyl [8,16] show two different relaxation processes: the ordinary (Y relaxation and an additional fast process, with relaxation time of the order of 3ps (in the range 250-450K). Similar results were obtained using NMR [lo]. It is very tempting to identify these experimental findings with the feature obtained here in the frequency range lo-100 GHz in the susceptibility.

We have also performed some test calculations at 205 K and at 266 K using a considerably larger system (2592 molecules, 7776 sites) in order to clarify the origin of the small oscillations around 3-8 ps. We find that these are reduced in magnitude and, more importantly, they are shifted to larger times (about a factor 2) [15], which shows that the oscillations are an artifact due to the periodic boundary conditions. Besides this, the temporal dependence is similar for the two system sizes but the excess relaxation is somewhat more pronounced for the larger system.

Fig. 4 shows the coherent part F(q, t) at T = 275 K (the supercooled region) at several different wavevectors around and below the first peak in the static structu:e factor. (For site-site correlations the first peak is located at qO =

1.5 A [15].) We can clearly identify the collective density oscillations, or quasi-lattice vibrations, as the oscillations for small q vectors. They are dispersive (the frequency increases1 for larger q-vectors), and are damped auf, being only visible for q < 1.0 8, . For q-vectors larger than qO = 1.5 A F(q, t) and F”(q, t) show very similar behaviour which indicates that the decay of the scattering function for these wavevectors is dominated by the single particle motion. From this we conclude that the quasi-lattice vibrations are well separated from the motions that are responsible for the anomalous decrease of the Debye-Waller factor.

Next we investigate if the above excess relaxation is mainly caused by


q.o.20 l/A - q.o.40 l/A ----. q=O.SO I/A ....’ q=, ,20 ,,A

q=1.60 l/A -.-.-

Fig. 3

d 0.1 1 10 100 Iwo v 0.1 1 10 100 loo0

frequency (GHz) time (ps)

Fig. 3. Fig. 4.

The susceptibility x”(q, w) = oS’(q, o) versus frequency for wavevector q = 1.94 A-’ in a log-log scale. The dashed lines shows the combined contribution from the microscopic motion (defined by processes faster than 3 ps) and the a relaxation (the stretched exponential).

Fig. 4. The intermediate scattering function F(q, t)lFQ, t = 0) at T = 275 K and for several different wavevectors. For small wavevectors q < 1.0 8, the quasi-lattice vibrations are clearly visible.

translational or rotational degrees of freedom. Our model for o-terphenyl is rigid and no intramolecular motions exists. Still, the molecule can rotate and to study the translational and rotational degrees of freedom we write the position of the site a (with R = 1, 2 or 3) in molecule i as R:(t) = RF”(t) + u:(t). We have then determined the correlations for the center-of-mass motion Rrm(t) and for the vectors u:(t), describing the orientation of the molecule. For the orientational correlation function we have made an average over the three different sites [15]. We find that the temperature dependence of the decay rate 7 = T(q, T) is very similar for the translational and orientational correlation functions for T > 275 K. The orientational correlation function show somewhat more stretching (p,, = 0.69 (for q = 1.94A-*) compared with &, = 0.82 for the center-of-mass motion, where p,, is the average value for the exponent /3 in the stretched exponential; the corresponding value for the site-site correlations is p,, = 0.74).

We have also investigated the molecular origin of the mesoscopic relaxation processes by determining the van Hove self-correlation function [ll], which reveal the details of the single particle motion in real space. The function has a single peak which moves to larger distances with increasing time; at lower temperatures, it is essentially arrested at about 0.7 A. There is no sign, at any temperature, of the ubiquitous two-peak structure found in simple atomic

G. Wahnstriim, L.J. Lewis I Molecular dynamics simulation of molecular glass 155

models in the supercooled regime [17]; the occurrence of a second peak is the clear signature of jump-diffusive motion with one or several particles involved. Clearly, the added complexity of molecular - as opposed to atomic - entities is sufficient to inhibit jump diffusion, at least on the nanosecond time scale under consideration here. At mesoscopic times, 2-20 ps, and for temperatures in the range 270-320 K, all molecules are still trapped within their cages formed by their neighbours. The motion is, however, strongly anharmonic, which shows up in the anomalous decrease of the Debye-Waller factor and in a strong non-Gaussian spatial dependence for the van Hove self-correlation function

1111. In conclusion, we have shown that our simple model for o-terphenyl

reproduces remarkably well the anomalous decrease of the Debye-Waller factor seen in inelastic neutron scattering experiments [2]. The MD data show that this results from strongly hindered motion of molecules temporarily trapped in the cage formed by the surrounding molecules.

This work is supported by grants from the Natural Science and Engineering Research Council (NSERC) of Canada and the “Fonds pour la formation de chercheurs et l’aide a la recherche” of the Province of Quebec to LJL, as well as by the Swedish Natural Science Research Council (NFR) to GW. The calculations reported here were performed on the Cray Y-MP at the Pittsburgh Supercomputer Center, thanks to a grant allocated to LJL by the “Services informatiques de 1’UniversitC de Montreal”.

References

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[2] E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu and W. Petry, Ber. Bunsenges. Phys. Chem. 93 (1989) 1252.

[3] W. Petry, E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu and B. Farago, Z. Phys. B 83 (1991) 175.

[4] M. Kiebel, E. Bartsch, 0. Debus, F. Fujara, W. Petty and H. Sillescu, Phys. Rev. B 45 (1992) 10301.

[5] G.P. Johari and M. Goldstein, J. Chem. Phys. 53 (1970) 2372. [6] L. Wu and S.R. Nagel, Phys. Rev. B 46 (1992) 11198. [7] E. Rossler, Phys. Rev. Lett. 69 (1992) 1620. [8] E.W. Fischer, G. Meier, T. Rabenau, A. Patkowski, W. Steffen and W. Thonnes, J.

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[lo] W. Schnauss, F. Fujara and H. Sillescu, J. Chem. Phys. 97 (1992) 1378. [ll] L.J. Lewis and G. Wahnstrom, Solid State Commun. 86 (1993) 295. [12] G. Wahnstrom, Phys. Rev. A 44 (1991) 3752.


[13] W. G&e and L. Sjiigren, Rep. Prog. Phys. 55 (1992) 241. [14] U. Buchenau, N. Nicker, A.J. Dianoux and D. Krause, Physica A 201 (1993) 372, these

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Molecular dynamics simulation of a molecular glass at intermediate times