24480
J. Phys. Chem. B 2005, 109, 24480-24488
Orientational Dynamics in the Isotropic Phase of a Calamitic Liquid-Crystal Model Davide Bertolini† Istituto per i Processi Chimico-Fisici, CNR, Via G. Moruzzi 1, I-56124 Pisa, Italy
Giorgio Cinacchi,* Luca De Gaetani,‡ and Alessandro Tani§ Dipartimento di Chimica, UniVersita` di Pisa, Via Risorgimento 35, I-56126, Pisa, Italy ReceiVed: June 20, 2005; In Final Form: October 17, 2005
We report a molecular dynamics simulation study on the isotropic phase of an idealized calamitic liquid crystal model with a length-to-width ratio of =5-6. The study focuses on the characterization of singleparticle and collective orientational dynamics on approaching the phase transition to the nematic phase. Recent experimental and simulation works have suggested that a power law behavior exists at relatively short times in the decay of the time derivative of the orientational correlation functions. Qualitatively, our simulation data are consistent with these findings. Both single-particle and collective time correlation function derivatives possess, in their respective log-log plots, a linear region at very short times, whose slope is essentially independent from the thermodynamic state. Nevertheless, the single-particle orientational correlation functions are better described by a function which is the sum of a fast exponential, an intermediate stretched-exponential and a slow exponential, while the collective orientational correlation functions are satisfactorily described by a sum of two exponentials, at higher density, or by just one exponential, at lower density.
1. Introduction The isotropic (I) to nematic (N) phase transition is experimentally known to be weakly first order.1,2 It has appeared natural to believe that the short-range structure of nematogenic fluids changes smoothly across the above-mentioned phase transition. This approximation, at the heart of the LebwohlLasher (LL) lattice model3 and its variants,4 has received a recent clear confirmation by the calculation of the direct correlation function (DCF) in a system of modified Gay-Berne (GB) particles: in fact, the DCF was found to remain short-ranged and weakly affected by the approach to the IN phase transition (INPT).5 The LL model, originally devised to test Maier-Saupe mean field theory for nematics,6 was later demonstrated to provide a good account of the peculiarities of the INPT, including the presence of pretransitional phenomena.7,8 The latter, observed experimentally, are a consequence of the weak character of the IN phase transition and were predicted by De Gennes, who extended the Landau theory of second-order phase transitions to the INPT.1 According to Landau-De Gennes (LDG) theory, the isotropic phase of a nematogenic fluid is seen as composed of pseudonematic domains, whose characteristic dimension, l(T), grows with decreasing temperature as follows:
[T -T*T*]
l(T) ) l0
1/2
(1)
where l0 is approximately the molecular length, and T* a temperature usually ∼1-2 K lower than TIN, the temperature at which the INPT occurs. LDG theory predicts, in addition, * E-mail: † E-mail: ‡ E-mail: § E-mail:
[email protected].
[email protected].
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that the orientation of the pseudonematic domains relaxes exponentially with a characteristic time given by
τLDG∝
η(T) (T - T*)
(2)
where η(T) is the shear viscosity. Equations 1 and 2 have received several experimental confirmations (see refs 9-11 and references therein), provided T is not too far from T*. Slightly above T*, pseudonematic domains are present, and relatively large and long-lived; on increasing temperature, their dimensions become progressively smaller, and it is natural to believe that they form, disrupt and reform with an increasingly larger frequency. At sufficiently large temperatures, l is comparable to the molecular dimensions: the domains no longer exist and the orientation of a molecule should relax according to the Debye law:
τDEB∝
η(T) T
(3)
The structure and interdomain orientational dynamics in the isotropic phase of a nematogenic fluid approaching the INPT are reasonably well understood. The interest in the intradomain dynamics has been recently renewed by the experiments described in refs 9-14, all seemingly indicating the existence of a temporal power law in the short-time decay of the optical Kerr effect (OKE) experimental signal. The latter should be essentially proportional to the time derivative of the collective orientational correlation function. Early experiments were believed indicative of an universal value of the exponent of the power law component, equal to -0.63;9-11 a theory was then developed to account for this behavior.15 Later experiments,12-14 however, have suggested that the exponent should be related to the effective length-to-width ratio (κ) of the nematogenic
10.1021/jp0533067 CCC: $30.25 © 2005 American Chemical Society Published on Web 12/03/2005
Orientational Dynamics
J. Phys. Chem. B, Vol. 109, No. 51, 2005 24481 TABLE 1: Thermodynamic Properties of the Four State Points Investigated state point
pressure (kbar)
temperature (K)
density (1000 Å-3)
I II III IV
2.48 2.56 2.57 2.51
598 623 636 632
2.076 2.070 2.053 2.040
Figure 1. Calamitic liquid-crystal model adopted.
molecules. The new theoretical analysis12 has led to an expression of the collective orientational correlation function valid for intermediate times, and such that its derivative decays as a power law in an appropriate short-time limit. The relaxation of single-particle and collective orientational correlation functions has been also the subject of a few computer simulation studies.5,16-19 Computer simulations are a very valuable tool that provides an effective bridge between theory and experiments, particularly when performed on simplified models. Early computer simulation studies were able to observe a significant slowing down of the collective orientational relaxation near the INPT.16,17 The decay of both single-particle and collective functions was found to be substantially exponential,16,17 even though the predictions of the Debye smallstep rotational diffusion model deteriorate as density is increased.17 Particularly interesting are the results of ref 5, since they anticipate, at a qualitative level, the essence of later experimental outcomes: the decay of the Fourier transform of the collective orientational correlation function was found to be exponential for not too large wavenumber, and the approach to the INPT affects only the longest wavelength dynamical properties. Power law decay in the single-particle second rank orientational correlation functions was observed in refs 18 and 19, where also second rank collective orientational correlation functions have been calculated. The power law component seems to emerge in the close vicinity of the INPT for both single-particle and collective functions. All numerical studies performed so far have adopted uniaxial ellipsoids, either hard or of the GB type, and with κ e 3, as models for the nematogenic molecules. However, it is known that shape has a profound influence on liquid-crystalline properties.20 Therefore, we have thought it worthwhile to extend the above-mentioned calculations to another calamitic liquidcrystal model, namely an array of nine soft spheres arranged linearly so that the overall particle resembles a spherocylinder of κ = 5-6. In refs 21 and 22, the phase behavior, structure, diffusion and viscosity of an ensemble of 600 such particles have been recently investigated. The present work focuses on the orientational dynamics of the same system in the isotropic phase. Molecular dynamics (MD) simulations in the microcanonical (NVE) statistical ensemble have been performed close to the INPT. Details about these calculations are provided in the next section, while the results are presented and discussed in section IV. Conclusions are finally drawn in section V. 2. Details on Model, Molecular Dynamics Simulations, and Computed Quantities The mesogenic particles considered are depicted in Figure 1. They are formed by nine soft spherical beads, held rigidly on a line. Contiguous beads are kept at a distance of 2.34 Å, while any pair of sites belonging to different particles interact according to the reference soft repulsive potential in the WeeksChandler-Andersen separation of the Lennard-Jones potential, with σ0 ) 3.9 Å and 0 ) 6 × 10-22 J.21,22 Only the two extreme sites bear a mass m ) 15 × 1.67 × 10-24 g.22 MD-NVE simulations (see ref 22 for a discussion on why the microcanonical ensemble has to be used if the interest is in calculating dynamical quantities) on a system of such rods, with N ) 600, have been performed at the state points in the isotropic
phase reported in Table 1. The equation of motions have been integrated by the Verlet algorithm with a time-step of 5 fs and the method of constraints has been employed to keep the particle linear and the constituent beads at the pre-selected distance (see the appropriate references cited in refs 21 and 22). In the present work, the interest is in the calculation of secondrank single-particle and collective orientational correlation functions. They are, respectively, defined as
C2s(t) ) 〈P2(uˆ i(t)‚uˆ i(0))〉
(4)
and N
C2c(t) )
〈P2[uˆ j(t) ‚ uˆ i(0))〉 ∑ j)1
(5)
where 〈〉 mean an average over particles i and time origins, P2() is the second-rank Legendre polynomial, and uˆ i(t) is the unit vector defining the orientation of particle i at time t. Since the calculation of any collective quantities requires quite long runs to accumulate averages with an acceptable statistics, the production runs lasted 20 ns at least, after a 1 ns run of equilibration in an isothermal-isobaric ensemble (see the appropriate references cited in refs 21 and 22). This length of the production runs results in a reasonable compromise between accuracy of the collective functions and computational effort. 3. Results and Discussion The signal detected in the OKE experiments should be essentially proportional to C˙ 2c(t), the time derivative of the collective orientational correlation function defined in eq 2. The experiments described in refs 9-14 span a time window approximately between 1 ps to 100 ns. The signal of the most recent experiments12-14 has been successfully fitted over the entire time window with the following function:
f (t) ) e-t/τLDG [a + e-t/γ(b + ct-p)]
(6)
The first exponential term describes the decay at long times (e.g., >1 ns for real substances). This relaxation is governed by the quantity τLDG, that is given by eq 2. This term is thought to describe the relaxation of pseudonematic domains that exist in the isotropic phase of a nematogenic fluid in the vicinity of the INPT. The terms in the square brackets, instead, are supposed to account for the intradomain relaxation, occurring on a shorter time scale (e.g., >1 ps and
