976
J. Phys. Chem. B 2006, 110, 976-987
Why Does the Intermolecular Dynamics of Liquid Biphenyl so Closely Resemble that of Liquid Benzene? Molecular Dynamics Simulation of the Optical-Kerr-Effect Spectra Guohua Tao and Richard M. Stratt* Department of Chemistry, Brown UniVersity, ProVidence, Rhode Island 02912 ReceiVed: October 14, 2005; In Final Form: NoVember 22, 2005
The combination of optical-Kerr-effect (OKE) spectroscopy and molecular dynamics simulations has provided us with a newfound ability to delve into the librational dynamics of liquids, revealing, in the process, some surprising commonalities among aromatic liquids. Benzene and biphenyl, for example, have remarkably similar OKE spectra despite marked differences in their shapes, sizes, and moments of inertiasand even more chemically distinct aromatics tend to have noticeable similarities in their spectra. We explore this universality by using a molecular dynamics simulation to investigate the librational dynamics of molten biphenyl and to predict its OKE spectrum, comparing the results with our previous calculations for liquid benzene. We suggest that the impressive level of quantitative agreement between these two liquids is largely a reflection of the fact that librations in these and other aromatic liquids act as torsional oscillations with oscillator frequencies selected from the liquid’s librational bands. Since these bands are centered about the librational Einstein frequencies, the quantitative similarities between the liquids are essentially reflections of the near identities of their Einstein frequencies. Why then are the Einstein frequencies themselves so insensitive to molecular details? We show that, for nearly planar molecules, mean-square torques and moments of inertia tend to scale with molecular dimensions in much the same way. We demonstrate that this near cancellation provides both a quantitative explanation of the close relationship between benzene and biphenyl and a more general perspective on the similarities seen in the ultrafast dynamics of aromatic liquids.
I. Introduction The window that optical-Kerr-effect (OKE) spectroscopy1 is beginning to provide into the intermolecular dynamics of complex liquids has made it natural to revisit some of the more fundamental issues underlying picosecond and sub-picosecond liquid-state dynamics. It has, for example, long been appreciated that the structure of neat, simple liquids is governed far more by the way in which the repulsive forces convey the molecular shape than by any specific attractive interactions2smaking it logical to assume that the ultrafast intermolecular dynamics ought to obey similar strictures. Yet, when benzene, itself one of the most famous examples of a repulsive-force dominated liquid structure,3 was subjected to OKE scrutiny, its peculiar spectral shape seemed to suggest that the aromatic interactions played a critical role.4 Moreover, the similar-looking spectra seen in subsequent OKE measurements of other, presumably simple, aromatic liquids appeared to lend added support to this notion.5-14 In a recent paper,15 we suggested that what one really needed to do to confront this issue was to carry out fully molecular level calculations of OKE spectra, ones that would allow us to analyze the spectra in microscopic detail. Using appropriate statistical mechanical techniques, it is now possible not only to simulate these spectra, but to assign their features to well-defined kinds of molecular motions.13,15-26 Indeed, we found that we were able to understand the experimental OKE spectrum of liquid benzene in some detail by using a simulation with an intermolecular potential devoid of any specifically aromatic features. The key to thinking about benzene’s spectrum, and * Corresponding author.
we suggested, to those of related aromatic liquids as well, was to focus on the similar rotational kinematics of these moleculess all of which, not coincidentally, were planar.15 What our (instantaneous-normal-mode-based) analysis told us was that almost all of the dynamics an OKE experiment sees in benzene arise from librational motion about the two in-plane axes, the high-frequency response reflecting largely singlemolecule librations and the low-frequency response coming from a number of different kinds of collective outer-shell motion coupled to these librations.15 Given that, the appearance of the spectra cannot help but be a direct consequence of the bandshape of these rotational components of the intermolecular dynamics. But, what makes planar molecules unique in this regard is that they inevitably have rotational bandwidths significantly broader than their translational counterparts. The origin of this behavior is fairly easy to understand. The kinematics of planar molecules is dominated by the fact that the moment of inertia for rotation about an axis perpendicular to the plane (the 6-fold axis in benzene) is always equal to the sum of the other two (identical, in the case of benzene) in-plane-axis moments of inertia. Planar molecules are, thus, always guaranteed to have one or two relatively small moments of inertia, ensuring relatively high librational frequencies and, thus, an extended rotational bandwidth.5,15,27 The ability of these ideas to explain why the nondiffusive portions (“reduced spectral densities”) of the OKE spectra of aromatic liquids such as benzene,4,5,7,12-15,28 aniline (C6H5NH2),9-11 and benzonitrile (C6H5CN)5,6,11 are so similar to one othersand so different from the corresponding spectra of nonaromatic (and nonplanar) species such as cyclohexane (C6H12),6 carbon disulfide (CS2),21,26,28-32 and acetonitrile
10.1021/jp0558932 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/21/2005
Molecular Dynamics Simulation of the OKE Spectra (CH3CN)16,17,19,29,33,34sand to do so without invoking any specialized intermolecular forces seems reassuring. However shortly after this work was published, Quitevis’ group pointed out that the OKE reduced spectral density of molten biphenyl (C6H5-C6H5) is virtually indistinguishable from that of benzene.35 The problem is that, at least at first glance, biphenyl seems to have nothing in common with benzene besides aromaticity. Biphenyl is not restricted to planarity in the liquid,36,37 and its much larger moments of inertia seem rather unlikely to allow for any quantitative similarities with benzene’s rotational dynamics. So, should one still say that the OKE spectrum is largely driven by the rotational kinematics of individual molecules in a bath? In this paper, we explore the generality of this kind of singlemolecule rotational-kinematics analysis by looking explicitly at the intermolecular dynamics and OKE spectra of molten biphenyl. What we find is that, surprisingly, the liquid-phase kinematics of biphenyl is remarkably similar to that of benzenes and that the same kind of OKE calculation that we carried out for benzene (one with no specialized aromatic forces) does indeed produce a biphenyl spectrum virtually identical to that of benzene’s. The reasons for this similarity, we suggest, lie in a universality of librational kinematics in simple liquids that is much more general than the relationship between benzene and biphenyl. Much of librational dynamics in ordinary, aprotic solvents can apparently be thought of as a kind of torsional oscillation occurring with a frequency on the order of the rotational Einstein frequency.38-40 Since these Einstein frequencies involve both the mean-square torque felt about a given rotational axis and the moment of inertia about that axis, one might have expected librational time scales to be rather molecule-specific. But while the specific details of a molecule’s geometry do control the torques felt at each of its atomic site as well as each site’s contributions to the moment of inertia, these two properties tend to scale with the molecular shape in very similar ways. The end result, as we illustrate with benzene and biphenyl but argue is much more common, is that such molecular details can largely cancel. We find that liquids of nearly planar molecules, in particular, should be remarkably similar to one another in their librational dynamics. We begin our exploration in this paper by simply looking at the results of a molecular dynamics (MD) simulation of liquid biphenyl. To ensure that we can make a fair comparison with our earlier study of liquid benzene,15 we adopt intermolecular potentials and a polarizability model that are natural generalizations of the reasonable, but not optimum, choices we made for benzene. We also restrict ourselves to completely rigid biphenyl molecules in which the ring-ring dihedral angle Φ is kept fixed. While in real liquid biphenyl this angle fluctuates about a value on the order of 30°,36,37 the opportunity to study the dependence of the intermolecular dynamics on Φ will turn out to provide us with a valuable tool for testing our picture of the kinematics. The remainder of this paper will be organized as follows: After summarizing our models and methods in section II, we present our results for the liquid structure, intermolecular dynamics, and OKE spectroscopy of biphenyl in section III, emphasizing the quantitative similarities with liquid benzene. We then show, in section IV, how a simple model for the distribution of torques at individual molecular sites allows us to predict the existence of some fairly universal relationships between librational Einstein frequencies for different liquidss and, in particular, allows us to understand the similarities we
J. Phys. Chem. B, Vol. 110, No. 2, 2006 977
Figure 1. Geometries and molecular-frame (center-of-mass) coordinate systems assumed for biphenyl (left) and benzene (right). In both cases, the axes are chosen so as to diagonalize the moment-of-inertia tensor and so as to make the x-axis a symmetry axis for 2-fold rotation about the phenyl groups. For benzene and planar conformations of biphenyl, the y-axis lies within the molecular plane; for more general choices of the biphenyl ring-ring dihedral angle, Φ, the y-axis bisects the two ring planes.
see between the Einstein frequencies of some of the benzene and biphenyl axes. We conclude in section V with a few general remarks. II. Models and Methods A. Force Laws and Simulation Details. The geometries and molecular-frame coordinate systems we assume for benzene and biphenyl are illustrated in Figure 1 and described in detail in Table 1. While benzene can undergo two fundamentally different kinds of rotation (tumbling about the in-plane, x and y, axes and spinning about the perpendicular, z, axis), a biphenyl with an arbitrary value of the ring-ring dihedral angle, Φ, can undergo three different kinds of rotation. Clearly, a rotation about the long (x) axis of biphenyl is going to be qualitatively similar to a rotation about the x-axis of benzene (though the former has double the moment of inertia). Rotation about the two perpendicular (y and z) axes of biphenyl, though, will have anywhere from 10 to 12 times the moment of inertia of benzene’s tumbling motion. The actual moments of inertia for the three values of the dihedral angle we study in this paper are listed in Table 2. The differing mass distributions notwithstanding, one might expect the intermolecular forces in the two nondipolar aromatic liquids to be reasonably similar. It therefore seems sensible to take for biphenyl’s intermolecular potential the same Williams form,41 with the same parameters, as the ones we used for liquid benzene: each carbon and hydrogen defines a site, with the site-site potential between site a on molecule j and site b on molecule k given by
[
uab(rja,kb) ) Bab e-Cabr -
Aab r
6
+
]
q aqb r
(2.1)
r)rja,kb
The parameters relevant for each kind of site are listed in Table 1.42 As noted in our earlier work15 (and Elola and Ladanyi observed in their complementary study of C6H6/C6F6 mixtures),43 while the Williams potential does a nice job of reproducing the structure and thermodynamics of liquid benzene, it is probably a bit stiffer than we would prefer for comparing with experimental dynamics. The full-width-at-half-maximum of the predicted reduced OKE spectrum of benzene is about 25% broader than the reduced experimental spectrum, for example. Nonetheless, since the Williams potential explains the shape of benzene’s spectrum reasonably well and since we want to emphasize the relationship between our predictions for biphenyl and our previous calculations for benzene, we shall continue to make use of it here. As with our benzene studies, our molecular dynamics simulations of liquid biphenyl were NVE calculations carried out with the MOLDY program44 using 108 molecules subject
978 J. Phys. Chem. B, Vol. 110, No. 2, 2006
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TABLE 1: Benzene and Biphenyl: Molecular Geometries and Interactionsa bond lengths (Å)
C-C (ring)
C-H
C-C (bridge)
benzene biphenyl
1.393 1.393
1.027 1.027
1.493
pair potential parametersb
C-C
H-H
C-H
A (kJ Å-6 mol-1) B (kJ mol-1) C (Å-1) q (au)
2439.8 369743 3.60
136.4 11971 3.74
576.9 66530 3.67
C (ring)
C (bridge)
H
-0.153
0.000
0.153
a
Benzene is described by a 12-site model consisting of a rigid hexagonal ring of C-H groups. Biphenyl is represented by a 22-site model consisting of two rigid hexagonal rings of C-H groups joined by removing one H from each ring and attaching the two (bridge) C atoms. The angle Φ between the two rings (defined to be the smaller of two angles between the ring planes) is held fixed. b Parameters for the Williams potential, eq 2.1, used for both benzene and biphenyl (refs 41 and 42).
TABLE 2: Principal Moments of Inertia for Benzene and Biphenyla benzene
Ixx
Iyy
Izz
87.63
87.63
175.26
biphenyl
Ixx
Iyy
Izz
Φ ) 0° Φ ) 30° Φ ) 90°
175.26 175.26 175.26
890.22 901.96 977.85
1065.48 1053.74 977.85
a Values in amu Å2 computed using the geometries described in Figure 1 and Table 1. The results for biphenyl are reported as a function of the dihedral angle Φ between the ring planes.
to periodic boundary conditions. Both sets of simulations used Ewald summations for the electrostatic interactions, while cutting off the nonelectrostatic interactions at half the length of the simulation cell.45 For values of the dihedral angle other than 0° or 90°, rigid biphenyl molecules are chiral, so we were careful to perform our studies of 30° biphenyl with a racemic mixture composed of 54 molecules of each handedness. We also chose to reduce the time step for all of our biphenyl simulations to 2 fs. In keeping with the basic goals of this study, we needed to ensure that we were examining comparable liquid conditions in these two substances. We therefore chose to look at biphenyl at a density of 0.9914 g/cm3 and at a temperature of 348 K, just above its experimental melting point (342 K). This temperature is somewhat higher than the 300K we used to study benzene (which melts at 278 K), but experimental OKE measurements reveal little dependence on temperature in the liquid range.35 However, cognizant of the fact that our planar (Φ ) 0°) biphenyl might be more prone to crystallize than the real biphenyl, we also carried out simulations at 393 K (at 0.953 g/cm3).46 For both of these thermodynamic conditions, we equilibrated our sample by melting an initially fcc lattice. The lattice was held at 1000 K for 1 ps, the kinetic energy was slowly rescaled to its desired final value over the next 49 ps, and the system was allowed to equilibrate for a subsequent 100 ps. Data were then sampled every 10 time steps throughout the observational time window. The angular velocity correlation functions we report were averaged over 9.9 × 103 such statistically independent liquid configurations while the mean-square torques were averaged over 2 × 104 configurations. The final OKE spectra required somewhat more extensive averaging: 10 1.2 ns trajectories were used to provide sample sets of roughly 6 × 105 configurations for each spectrum. B. Optical-Kerr-Effect Spectra. The OKE spectra themselves are computed from the time evolution of the liquid’s many-body polarizability tensor Π 6 . In isotropic liquids, it is
convenient to calculate the time-domain response function R(3)(t) for an OKE spectrum
R(3)(t) ) -β
{
}
d 1 1 〈PP[Π 6 (t)Π 6 (0)]〉 - 〈Tr[Π 6 (t)]Tr[Π 6 (0)]〉 dt 10 30 (2.2)
from tensor invariants rather than individual tensor elements.23,47 Here, for any two tensors 6 A and 6 B, we define
PP[A 66 B] ≡
∑
AµνBµν, Tr[A 6] ≡
µ,ν)x,y,z
∑
Aµµ (2.3)
µ)x,y,z
The angular brackets 〈...〉 in eq 2.2 represent ensemble averages and β ≡ (kBT)-1. The corresponding frequency-domain spectrum is given by
Im[R(3)(ω)] )
∫0∞R(3)(t) sin ωt dt
(2.4)
Since we are interested most in the nondiffusive portion of the intermolecular dynamics (the reduced spectral density), we need to remove the long-time rotational-diffusion contributions from eq 2.2. As we have emphasized in our earlier work,15 this transformation is not simply an empirical device; one could, in fact, calculate these contributions exactly by simulating the asymptotic behavior of appropriate orientational correlation functions. However, a simpler (and sufficiently accurate) approach is just to fit the long-time decay of the OKE response function to a sum of exponentials -t/τr )(A2e-t/τ2 + A3e-t/τ3) R(3) diff(t) ∼ (1 - e
(2.5)
and to compute the reduced spectral density Im[R′(ω)] as the imaginary part of the difference between this function and the full response function
R′(t) ) R(3)(t) - R(3) diff(t) Im[R′(ω)] )
∫0∞R′(t) sin ωt dt
(2.6)
(In practice, the parameters in the second parenthesis in eq 2.5 are obtained by fitting the full response function for times longer than 0.5 ps to an expression of the form 3
R(3)(t) ∼
Aie-t/τ , ∑ i)1 i
(τ3 > τ2 > τ1)
and by discarding the presumably nondiffusive subpicosecond τ1 term. The results are not particularly sensitive to the rise time τr, so we simply use the standard experimental procedure35 to assign it a plausible value capable of representing the nonin-
Molecular Dynamics Simulation of the OKE Spectra
J. Phys. Chem. B, Vol. 110, No. 2, 2006 979
TABLE 3: Diffusive Contributions to the OKE Spectrum of Biphenyla 348 K Ai b (ns-1)
τi (ps)
Ai b (ns-1)
τi (ps)
10.88 8.04
0.270 4.534 43.77
26.81 22.50
0.302 3.437 21.60
onsetc
i)2 i)3
393 K
a Amplitude (Ai) and lifetime (τi) are parameters in eq 2.5 calculated from simulations of the OKE spectra of liquid biphenyl at 348 K (0.9914 g/cm3) and 393 K (0.953 g/cm3) with a ring-ring dihedral angle of 30°. b The asymptotic form of the response function is expected to scale with βNγ2/15, with N being the number of molecules, β ) (kBT)-1, and γ being the polarizability anisotropy given in Table 4 (ref 15), so the amplitudes reported here are for the simulated response functions divided by this prefactor. c The lifetime listed under “onset” is the rise time τr, which we estimate in the standard fashion (see ref 35) as (2〈ν〉)-1, with 〈ν〉 ) (∫dω (ω/2π) Im R(3)(ω))/(∫dω Im R(3)(ω)), the mean frequency of the (nonreduced) OKE spectral density. We find 〈ω/2πc〉 ) 61.84 cm-1 at 348 K and 55.20 cm-1 at 393 K.
stantaneous character of the rise of the diffusive contribution from zero. The parameters we use are reported in Table 3.) The many-body polarizability required for these calculations can be modeled at a number of different levels.18,19 Our benzene study included all of the infinite order dipole-induced-dipole contributions, making the standard (and somewhat simplistic) approximation that each molecule can be described with single point-dipole polarizability located at the center of mass.15 Biphenyl, though, is sufficiently large that failing to distribute the polarizability contributions throughout each molecule would be likely to be rather unrealistic.48 To keep as many parallels as possible with our treatment of benzene, we therefore adopt a compromise two-site model for biphenyl,49 placing a pointdipole polarizability tensor identical to benzene’s at the center of each phenyl ring and incorporating all of the resulting sitesite intermolecular dipole-induced-dipole contributions in the liquid. For our N molecule liquid, then, we write the many-body polarizability as a sum over the liquid-state polarizabilities of each site p in each molecule j, 5 πp(j) N
Π 6)
2
∑ ∑5πp(j) j)1 p)1
(2.7)
TABLE 4: Polarizability Tensor for Biphenyla theoreticalb experimentalc
Rxx
Ryy
Rzz
R
γ
23.51 24.7
22.89 20.2
14.80 13.8
20.4 19.6
-8.42 -9.49
a Values in Å3 for Rxx, Ryy, and Rzz, the eigenvalues of the polarizability tensor of biphenyl, and for R and γ, the mean polarizability and polarizability anisotropy. b Computed for an isolated biphenyl molecule, assuming a fixed phenyl-phenyl dihedral angle Φ ) 30°, using the two-site model described in the text, eqs 2.10 and 2.11. c Derived from measurements of biphenyl dissolved in CCl4 (ref 52).
Figure 2. Center-of-mass radial distribution functions computed for liquid benzene (black) and for liquid biphenyl (colored), with two different choices of the biphenyl ring-ring dihedral angle (Φ ) 30° and 90°). For benzene, the simulation temperature and density are 300 K and 0.874 g/cm3. The two biphenyl simulations are carried out at 348 K and 0.9914 g/cm3.
effectiVe site polarizabilities in which the intramolecular sitesite interaction has already been taken into account.51 As can be seen from Table 4, the single-molecule polarizability implicit in this treatment 2
5(j) ) R
∑R5p(j)
(2.11)
p)1
actually turns out to be in reasonably good agreement with the measured polarizability tensor for biphenyl.52 III. The Structure and Dynamics of Liquid Biphenyl
N
5p(j) [1 6+ 5 πp(j) ) R
2
∑ ∑6T(rjp,kq)π5q(k)] k*j q)1
(2.8)
with p, q ) 1, 2 denoting the two phenyl rings, 6 T(rjp,kq) the dipole-dipole tensor between two phenyl rings on biphenyls j and k,
1 )/r3]r)rkq-rjp 6 T (rjp,kq) ) [(3rˆ rˆ - 6
(2.9)
and R 5p(j) the (isolated-molecule) polarizability tensor of the pth phenyl ring of the jth biphenyl molecule
(
5p(j) ) R R
γ 6 1+γΩ ˆ (j,p)Ω ˆ (j,p) 3
)
(2.10)
In this last expression, we take R ) 10.20 Å3 and γ ) -4.67 Å3, the values of the isotropic and anisotropic polarizabilities of benzene,50 and define Ω ˆ (j,p) to be an orientational unit vector describing the normal to the phenyl plane. Despite our use of benzene-like numbers in eq 2.10, these phenyl ring polarizabilities should probably be regarded as
A. Liquid Structures of Biphenyl and Benzene. From the perspective of their respective centers of mass, the radial distribution functions of liquid biphenyl and liquid benzene look rather different (Figure 2). Apart from the rising edges, which occur at nearly identical locations, the two liquid structures might seem to have little in common. However, a more interesting take on the structures comes from sitting in the center of each phenyl group and asking about the distribution of the surrounding phenyl group centers. The resulting radial distribution functions (Figure 3) now exhibit markedly similar length scales, with pronounced solvation shells clearly visible for phenyl-phenyl distances rPh-Ph between 4 and 8 Å and between 8 and 12 Å. Interestingly, the details of this biphenyl structures and the similarities with benzenesseem to be largely unaffected by the dihedral angle. The absolute number of phenyls in each shell around a biphenyl actually match the corresponding numbers in benzene reasonably precisely, regardless of the biphenyl dihedral angle (Table 5). To see how far these similarities extend, we need to look at the angular distribution of the phenyl groups as well.43 For example, if we put ourselves in the molecular frame of a phenyl
980 J. Phys. Chem. B, Vol. 110, No. 2, 2006
Tao and Stratt
Figure 3. Phenyl-ring-center/phenyl-ring-center radial distribution functions for liquid benzene (black) and liquid biphenyl (colored). The thermodynamic states and choices of dihedral angles are the same as those used in Figure 2. Note that both biphenyl liquids exhibit a spike at 4.279 Å corresponding to the fixed distance between intramolecular phenyl rings.
TABLE 5: Phenyl Group Distribution in Liquid Benzene and Liquid Biphenyla Ph-Ph distance 3-6 Å
6-8 Å
8-12 Å
7.68
35.5
8.84 9.32
40.0 39.9
Figure 4. Probability distribution of phenyl rings around a given phenyl ring in liquid biphenyl and in liquid benzene, plotted as a function of direction. These figures look at the phenyl rings whose ring-centerto-ring-center distance rPh-Ph falls within the range 3 Å < rPh-Ph < 6 Å and illustrates how the probability of finding the center of a neighboring phenyl varies with the spherical-polar angles θ and φ. The thermodynamic states are the same as those used in Figures 2 and 3. The biphenyl ring-ring dihedral angle Φ ) 30°.
Benzene 5.84 Biphenylb
Φ ) 30° Φ ) 90°
5.70 + 1 5.43 + 1
a The number of phenyl group centers within the indicated distance of a given phenyl group. The values for liquid benzene are for 300 K and 0.874 g/cm3; the values for liquid biphenyl are for 348 K and 0.9914 g/cm3 with ring-ring dihedral angles, Φ,fixed at the indicated values. b The “+ 1” listed under 3-6 Å refers to the adjoining intramolecular phenyl, which is always located at 4.279 Å; the number on the order of 5 is the number of phenyl groups found on different molecules.
group on the negative x side of a biphenyl (or in the molecular frame of a benzene molecule) and look within a prescribed range of rPh-Ph values, at which angles are we most likely to find another phenyl? The relevant distribution functions
g(θ,φ) (a < rPh-Ph < b) )
With a little further work, it is possible to prove that even the orientations of these phenyl neighbors are remarkably similar in the two liquids. If we weight our angular distribution by the cosine squared of the angle between the 6-fold axes of the phenyls, θPh-Ph, the resulting probability density (normalized by the likelihood of finding a phenyl in the prescribed direction)
cos2 θPh-Ph(θ,φ) (a < rPh-Ph < b) ≡ N
〈
cos2 θPh-Ph δ(θ - θifj)δ(φ - φifj)〉a
