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Molecular dynamics of discotics in the isotropic phase Arnold Maliniak Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden

(Received 9 September 199 1; accepted 2 1 October 199 1)

Molecular dynamics (MD) simulation and multifield carbon-13 nuclear spin relaxation data are reported and used in an investigation of dynamic processes taking place in the isotropic phase, at 95 ‘C, of benzene-hexa-n-heptanoate (BHA7). The relaxation data are analyzed by using a dynamic model based on separation between a fast internal and a slow overall motion. The overall reorientations defined by the motion of the rigid aromatic core, occurs on nanosecond time scale, whereas the internal motion in the flexible, aliphatic side chains is much faster, with the correlation times in the picosecond range. Increased mobility and decreased orientational order along the chains-characterized by order parameters-is predicted by the experiments and also confirmed in the simulation. The translational diffusion constants are calculated in the MD simulation and measured using FT-PGSE technique. The molecular structure obtained from the MD simulation indicates that the aliphatic chains are situated alternatively up and down relative to the aromatic core. The radial distribution functions show that the molecules in the liquid are associated in pairs, with a parallel orientation of the planes defined by the aromatic core.

I. INTRODUCTION

Studies of disklike molecules over the last years have attracted considerable attention in the scientific literature. Generally, diskotics consist of a rigid, usually aromatic core, to which several aliphatic chains are bonded in a symmetric way. Particularly, the mesomorphic properties of these molecules have been extensively investigated,14 but also the isotropic phase5 and the polymorphism in the solid state.6 Re- cently, a number of investigations of polymeric, diskotic liquid crystals has been reported.7-9

In the present paper a study of benzene-hexa-n-heptanoate (BHA7), Fig. 1, in the isotropic phase is presented. Hexa-n-alkanoates of benzene were the first disklike molecules with mesomorphic properties to be discoveredlOvll and BHA7 is the smallest member in this group, BHA6 homolog does not form liquid crystalline phases. The properties of BHA7 have been investigated by using different spectro- scopic methods: IR,12-14 ‘H,15 and i3C 16*17 NMR (nuclear magnetic resonance), as well as various calorimetric measurements.14,‘8 BHA7 exhibits only one liquid cristalline phase and therefore has a simple phase diagram:19

81.2’C 87.O'C

Solid + D, -+ Isotropic,

where D, denotes the mesophase with a tilted columnar structure, where the columns are forming a pseudohexa- gonal lattice. Several subjects have been addressed in the investigations of BHA7, such as the orientational order of the aromatic core in the liquid crystalline state, the disorder of aliphatic chains, and the thermodynamic functions associated with the phase transitions.

The current study, performed in the isotropic phase at 95 ‘C, is a combination of molecular dynamics (MD) simulation technique and carbon- 13 nuclear spin relaxation measurements on nuclei in the aliphatic chains. Molecular dy-

namics simulations have been previously performed on a large number of macromolecules, both in isotropic solu- tions20.21 and in oriented systems, such as liquid crystals22-25 and bilayer membranes. 26*27 The computer simulation

W13

lb)

FIG. 1. Molecular formula (a), and numbering system of benzene-hexa-n- heptanoate (BHA’I), C,(OCOC,H,,), (b).

2306 J. Chem. Phys. 96 (3), 1 February 1992 0021-9606/92/032306-12$06.00 @ 1992 American Institute of Physics

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method is powerful in combination with various experiments, since a single simulation provides information on both statical and dynamical parameters. Another important feature is that information on the molecular level is provided, in contrast to most of the experimental techniques that need a theoretical coupling between the macro- and mi- croscopic world. Carbon- 13 nuclear magnetic relaxation studies have been used extensively in investigations of dynamical processes taking place in macromolecules and ag- gregates. Many different models for analysis of relaxation data have been proposed, in particular the models where a fast, internal motion, of a segment in the molecule is superimposed on a slow overall reorientation. These models have been used in studies of biomolecules28-30 and isotropic, micellar solutions of surfactants.3’-33 In particular, the local dynamics of aliphatic chains attached to polar groups can be investigated and characterized in terms of an order parameter and a correlation time. The information about the structure and dynamics of the alkyl chains is essential for the understanding of many ordered systems, such as membranes and liquid crystals. The phase transitions of the mesomorphic systems can be correlated to a gradual melting of the aliphatic chains.

the simulation box, which was a compromise between a relatively long cutoff and the computer resources. The MD program that was used is based on simulation package McMol- dyn,36 the simulation was carried out on a CRAY XMP/416 and a Convex C220. All the CH, and CH, were treated as single interaction centers, which gives 60 sites per molecule and a total of 1920 interaction sites in the computational unit cell. During the equilibration the density of the system was increased from 0.7 g cm - 3 to the experimental value of 0.96 gy - 3,37 which gives the size of the cubic simulation box of 36 A. In the absence of crystal structure data, the initial conformation of the BHA7 molecule was obtained from an extensive molecular mechanics calculation using the MM2( 87) force field parameters.38 A large number of conformations was generated and the lowest energy conformation was used as the input to the simulation. During the initial 70 ps of the equilibration the molecules were kept rigid. The length of the production simulation was 100 ps where the molecules were flexible and were treated with atom based potential functions that can be separated in covalent (the first three terms), and nonbonded terms. The func- tional form of the potential is given by

In the present study of BHA7, attention is focused on the dynamical processes. Translational and reorientational dynamics are investigated, the latter includes the overall motion of the aromatic core and the segmental motions of the different alkyl groups in the aliphatic chains. In addition to the dynamics, the MD simulation also provides valuable information on the structure of the liquid and the conformations of the aliphatic chains.

E tOtal = C Kr(r--r,)2+ C JL(B-~&)~ bonds angles

V” +dibgralsy [l +cos(n9)l

+&((~y2-($y) +e] 3 (1)

In Sec. II A the method of simulation and the details of the model are described, whereas Sec. II B gives the background of the molecular reorientations and nuclear spin relaxation. In Sec. III the experimental procedures are discussed. The results from the simulation presenting static properties are described in Sec. IV A, finally, Sec. IV B is devoted to dynamic properties with subsections 1 and 2 in which, translational and reorientational motions are discussed, respectively.

II. GENERAL BACKGROUND A. Method and model of simulation

The molecular dynamics simulation was carried out on a system consisting of 32 molecules in a cubic cell. The equa- tions of motion were integrated using Verlet leap-frog algo- rithm with a time step of 0.001 ps. The periodic boundary conditions in three dimensions, together with minimum im- age convention and a nonbonded, spherical cutoff were applied on the center of mass of the molecules. The simulation was performed at a constant temperature (368 K) using constraints, and therefore essentially in a canonical ensem- ble (NVT). The constant temperature was achieved by res- caling the velocities at each MD time step.34,35 The reason for performing the simulation at a constant temperature, and not at a constant energy, was that initial short simulations showed large temperature fluctuations. These fluctuations can be explained by the small number of molecules in

where K,, Ke, and V,, are force constants representing bond stretching, bond bending, and torsional motion, respectively. The distance between the interacting sites i and j is rij whereas 6, o, and q are the parameters for the Lennard-Jones and electrostatic terms. The parameters describing the covalent interactions in the polar part of the molecule were taken from MM2(87) force field,38 whereas for the aliphatic chains, hexane parameters for the torsional angles,39 based on the same force field were adopted. The SHAKES algo- rithm was used to constrain the bonds’ lengths. The parameters for the covalent interactions are collected in Table I. The nonbonded part of the potential e.a. Lennard-Jones parameters, and the partial charges, are summarized in Table II. Standard combination rules are used to calculate the cross terms. All the nonbonded parameters were taken from the OPLS4’ force field. It can be noted that the united atoms are assumed to be neutral, q = 0. The scaling factors of 8.0 and 2.0 were used for the Lennard-Jones and electrostatic terms, respectively, for 1,4 nonbonded interactions. These factors4’ were originally introduced as a merger with the AMBER force field, but the stability of the system increased upon such scaling also for the present force field.

B. Molecular reorientations and nuclear spin relaxation The reorientations of a unit vector, ii, fixed in the molec-

ular frame can be described by the time correlation function

c(t) = (&[uN*fict,l), (2)

J. Chem. Phys., Vol. 96, No. 3,l February 1992

Arnold Maliniak: Discotics in the isotropic phase 2307


2306 Arnold Maliniak: Discotics in the isotropic phase

TABLE I. Covalent potential parameters. The atom types are defined in Fig. 1.

Bonds ‘4cA, K,(kJmol-‘A-‘)

CA -CA 1.34 2887. G-0, 1.36 1804. 01-G 1.34 1519. G-0, 1.20 3398. CO-C, 1.52 1323. C-c, 1.53 1087. C-G 1.53 1087.

Bond angles 0, (deg) &(Jmol-‘deg-*)

CA -CA -CA 120.0 30.93 CA-CA-O, 124.3 50.38 C*-o,-G 109.0 28.82 0, -G-a 122.0 57.61 0, -CO-c, 107.1 46.79 CO-C, -G 107.8 32.41 0,-&-c, 122.5 33.11 C”-c”-c” 112.4 35.48 c,-q-c, 112.4 35.48

Dihedral angles V, (kJ mol-‘) V,(kJmolV’) V,(kJmol-‘)

c,-c,-c,-c, - 3.88 CA-G-CA-O, 0.00 CA-G-O, -CO 0.00 G-0, -G-o* 0.00 c*-o,-G-c, 0.00 Q-CA-CA-O, - 8.36 0,-%-c, -G 1.67 co-c, -G-c, 2.09 02-ccJ-c,-G - 5.43 C”-c”-c”-c” 5.89 C”-G-c,-G 5.89

33.44 0.00 67.92 0.00

0.00 2.09 0.00 2.09 0.00 2.09

67.92 0.00 - 1.25 - 0.29

1.55 0.00 3.76 0.21

- 1.13 13.15 - 1.13 13.15

where the second order Legendre polynomial, PI (x) = l/2( 3x2 - 1 ), is relevant for nuclear spin relaxation times. The correlation functions provide detailed information on molecular reorientations, and can be calculated in molecular dynamics simulations. Particularly, if the unit vector, ii, is chosen along a C-H bond in a CH, group of an aliphatic chain of BHA7, the dynamic processes taking place in this segment, and affecting carbon-13 relaxation, can be investigated. The time correlation function itself cannot be obtained from the relaxation experiments, however, the spectral density function, J(o), that determines nuclear spin relaxation processes, is defined as the cosine Fourier transform of the time correlation function

TABLE II. Nonbonded interactions.

s

m

5(w) = 2 (cos wt) C( t)dt. (3) 0

The spin-lattice relaxation of the carbon-l 3 nuclei in the aliphatic chains of BHA7 is primarily due to dipolar interactions with the attached protons, and therefore the relaxation times, T,, and the nuclear Overhauser enhancement (NOE), 7, are given, for a segment i by”*

p”yHl;;h’2T) 1 @H -@,I CH

and (da)

vi= EL.

( ) YC

Site e,,/(kJ molt ‘) g,,(A) q,(e)

CA 0.460 3.750 0.30 0, 0.711 3.000 -0.40 CO 0.439 3.750 0.55 0, 0.878 2.960 - 0.45 C” 0.493 3.905 0.00 C, 0.493 3.905 0.00

X- 6&l, $-co,) -ao* --WC)

J(o, -wc) +3&o,) +&UH +%I ’ (4b)

where yH and yc are magnetogyric ratios of proton and carbon, rcH denotes the C-H bond length, and the other sym- bols have their usual meaning. Since the reduced spectral densities, j(w), contain the dynamical information about the system-through the time correlation function for the reorientation of a C-H vector-the interpretation of the re-

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laxation data implies an assumption regarding the form of the correlation function, or more generally, regarding the motional model for the reorientations of the vector. A model for interpretation of relaxation data by assuming a fast, local motion superimposed on a slow reorientation of a macromo- lecule or an aggregate has been proposed by Wennerstriim et aZ.43*44 It is usually termed “two-step” model referring to the motions occurring on two different time scales, with the fast motion being in the extreme narrowing region. Lipari and Szabo4” have proposed a “model free approach” which-in the case of isotropic overall motion and time scale separation between the slow and the fast motion-is equivalent with the two-step model. According to the model free approach, the reduced spectral density function for the anisotropic overall motion, valid in the time scale separation limit, is expressed as

r ASP, (1 --A)S27-, J(o)=2

1 . +

1 + (wr, )2 1 + (07-2 )’

I A(1 -S2)7’ -c ’ 1 + (UT’)* ’

(1 --A)(1 -S2)7”]. (5) l+(U8)* 1’ ”

where S is a generalized order parameter that reflects the restriction of the internal motion in a model independent way. For a completely rigid body S = 1, whereas isotropic internal motion gives S = 0. The correlation times 7, and r2 are related to the overall motion of the molecule, A (O<A ( 1) is the mixing parameter that describes the anisotropy of the overall motion and which can also be seen as an order parameter for the slow motion, and

(r’)-‘=rf’+r’-1, (r”)-‘=rJ-‘tr;‘,

(W (6b)

where ‘T/ is the correlation time for the fast motion, which in the case of BHA7 is the local motion of a segment in an aliphatic chain. In the case when 7’ = r2 = 75 or A = 1, the expression is recovered for isotropic overall reorientation and r, is the correlation time for this slow motion.

The relaxation times for the different carbons in the aliphatic chains of BHA7 show clearly a magnetic field de- pendence. In case of the anisotropic overall motion, the 13C relaxation time of each alkyl group is determined by five parameters ( r, , r2, TV, S, A ), whereas only three parameters are required in the isotropic case. Therefore, at least three independent measurements of spin-lattice relaxation times, T, , and/or NOE factors are necessary in order to extract all the dynamical parameters. In the present study the spin- lattice relaxation times are measured at three different magnetic fields and the NOES factors at two fields.

III. EXPERIMENTAL SECTION The compound, BHA7, was prepared by Zimmermann

at Max-Planck-Institut fiir Mediziniche Forschung, 6900 Heidelberg, Germany, according to a previously described46 procedure. The purity of the sample used in the present investigation was established by the sixfold symmetry of the molecule confirmed by the carbon- 13 NMR spectrum.

The carbon- 13 spin-lattice relaxation measurements were performed at three different magnetic fields: 4.7, 6.3,

and 9.4 T on a Bruker 200MSL, a JEOL GX270 and a JEOL GX400, respectively. The standard fast inversion recovery technique was used and the decoupler power was attenuated to avoid excessive heating of the sample. In the experiments at 4.7 T the WALTZ decoupler scheme4’ was used. The relaxation times were evaluated by three parameters, nonlinear fitting of the intensities and their accuracy is estimated at + 5%. The experiments were performed at least three times

and the reproducibility was within 5%. A large number of scans (typically 3000) was required due to a very small sample size, resulting in problems with the sensitivity. For the same reason we were not able to accurately determine the relaxation times for the methyl group, C,, and for the aromatic and carbonyl carbons, therefore these values are not reported here.

The steady-state NOE factors were determined at two magnetic fields of 4.7 and 9.4 T using standard technique. The peak intensities of the carbon signals were recorded with the proton decoupler turned on before and during acquisition, and compared with the intensities recorded with decou- pling only during the acquisition. In the experiments at 9.4 T the triple-resonance channel with the same power level but 100 kHz frequency offset was turned on during the pulse delay in order to achieve a constant heating of the sample. In analogy with the relaxation time measurements, in the experiments at 4.7 T the WALTZ decoupler scheme was used. The reported values are the averages of three experiments and the accuracy of the NOE factors is about f 0.1 unit. All the experiments were performed at 95 “C, the temperature was calibrated using ‘H chemical shift in ethylene glyco1,48 with an uncertainty of + 1 “C.

The translational diffusion coefficients were obtained from the Fourier transform pulsed-gradient spin-echo (FT- PGSE)49 measurements performed on a Bruker MSLlOO spectrometer operating at a magnetic field of 2.35 T. The diffusion coefficients were measured at 95 and 105 “C using the ‘H signal from the aliphatic chains (not resolved), and the reported values are averages of two measurements, with an error estimated at f 5%.

IV. RESULTS AND DISCUSSION A. Equilibrium and structural properties 1. Molecular structure

The time averaged intermolecular energies obtained from the simulation are given in Table III, together with simulation parameters for the system. It can be noted that

TABLE III. Simulation parameters and averages.

Number of interaction sites 1920 Density, g cm - 3 0.96 Cubic box length, A 36 Temperature, K 368 Time step, fs 1.0 Equilibration time, ps 100 Simulation length, ps 100 Intermolecular energy, kJ mol - ’ - 116.7 + 0.8 Electrostatic contrib., kJ mol - ’ - 18.7 f 0.5


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A.2

carbon - carbon

1.0 -

0.5 -

0.01 ’ I 1 ’ ’ ’ ’ 1 ’ ’ . 1 ’ . . 0 5 10 15 20

r(A)

FIG. 2. Radial distribution function for the aromatic carbons, g,,,,

the interactions are mainly determined by the Lennard- Jones part of the potential and the electrostatic contribution is of minor importance. Ab initio, quantum chemical calcula- tions5’ indicate that treating aliphatic chains as neutral is a reasonable approximation. In case of chains attached to a polar core, probably the o-carbon could be given a small partial charge.

The radial distribution function between the carbonyl oxygens, gozo2, is shown in Fig. 3. Since the aliphatic chains are arranged alternatively up and down relative to the aromatic ring, three different relative orientations of the car- bony1 groups, in two neighbor molecules, can be distin- guished. The first maximum at a distance of 3.0 A, corresponds to the carbonyl oxygens pointing towards each other. The second maximum at 5.7 A is related to the two carbonyl groups pointing in the same direction, both up or both down, relative to their rings. The third orientation, reflected in a peak at 6.3 A, corresponds to the carbonyl groups pointing in different directions, one up and one down. Final- ly, in analogy with the distribution function for the aromatic carbons, the broad maximum at 13 A corresponds to the oxygens in molecules at longer distances.

The radial distribution function for the methyl groups, gc,e,, is shown in Fig. 4. The only structure in this distribu-

The molecular structure obtained from the final configuration of the MD simulation shows clearly that the aliphatic chains of the molecule are alternatively situated up and down relative to the aromatic core plane. This structure was also found to be the most stable for ester chains attached to a triphenylene core, as derived from SCFKNDO calculations combined with x-ray diffraction,5’ and have also been discussed for BHA7 in solid state.16 A MM2 energy minimization performed on some molecules from the final MD configuration gave, on average, a total energy of - 6.6 kcal/mol. This can be compared with - 1.5 kcal/mol obtained from the corresponding minimization of the initial structure (before the MD simulation). It has been previously reported’2~53 that compared to molecular mechanics calculations, the MD simulations are much more efficient as a tool for determining conformational energy minima. The reason for this is that the fluctuations of energy in an MD

1.0 -

simulation can move the system from an energetic local minimum and pass an energy barrier, which is not possible in standard molecular mechanics. We will return to the discus- B sion on the molecular conformations in the next sections. 0.5 - 2. Radial distribution functions

The structure of the liquid is examined by using the radial distribution functions, g( ril ), where rii is the distance between sites i and j. In Fig. 2, radial distribution function for aromatic carbons, gc,,, , is shown. The first maximum, 5.5 A, corresponds to the distance of the energetically most stable configuration between two BHA7 molecules. The short distance indicates that the two aromatic rings are parallel,


this molecular arrangement is fundamental to all diskotic mesophases and has also been reported for the triphenylene compound in the crystalline state.51 It can also be noted that in the columnar mesophase of BHA7, the distance between the molecules, estimated from x-ray diffraction was 4.6 A.” Therefore, considering the difference in the density and in the molecular order between mesophase and isotropic liquid, 5.5 A seems to be a reasonable distance. The second broad maximum at 15 A is due to other molecules in the box with no specific relative orientation. The radial distribution function is a measure of local deviation from the bulk density, and should therefore, at long distances, approach unity. Apparently, this is not the case, which can be attributed to simulation artifact that is a consequence of applying the cutoff on the centre of mass instead of on the atomic sites. The sampling of the radial distribution functions was done during the simulation in the force calculation routine where the cutoff was applied on the center of mass in order not to split the molecules. It can also be noted that the radial distribution function suffers from statistical noise.

0.0-l . I . . / I I 0 5 10 15 20

r(A)

FIG. 3. Radial distribution function for the carbonyl oxygens, go+,,

J. Chem. Phys., Vol. 96, No. 3,i February 1992



methyl - methyl

5 10 15

r(A)

FIG. 4. Radial distribution function for the methyl groups, gcbce.

tion function consists of a strong, relatively broad maximum at 4.5 A due to neighbor methyl groups. However, these cannot be assigned to any specific molecules, the way it was done with the distribution in Figs. 2 and 3.

So far the intermolecular radial distribution functions were discussed. The intramolecular distribution functions are displayed in Fig. 5. These show the probability of finding a methylene or a methyl group, go+, within a chain, as a function of the distance from carbonyl oxygen. Note, that these functions differ from the distribution functions in the previous three figures by the way they have been normalized. In a standard radial distribution function, the number of particles is given by the volume integral ofg( r), in Fig. 5 the area enclosed by each function is constant and corresponds (after normalization) to one. The first distribution, n = 1,

o*4/ 2

22

: 12

g 0.2 'iZ 2 .- I; .o" n

0.0 L

x0.3

0

m-1

5 10 r(A)

FIG. 5. Intramolecular radial distribution function between the carbonyl oxygen and the segments, C,, in the same aliphatic chain. Arbitrary units.

shows as expected a single peak, essentially a S function, at 2.3 A. Note that the intensity of this distribution function is scaled by a factor of 0.3. Disregarding the bond vibrations and bendings, this distance is fixed. The second distribution, n = 2, shows two peaks with the relative intensities of 1:2, corresponding to the gauche and tram conformations. The tram conformation is the most populated one, and using Boltzmann statistics the tram-gauche energy difference was estimated to be 1.25 kJ/mol. This number is clearly smaller than corresponding energies in hydrocarbon chains, which are on the order of 3 kJ/mol.” A possible explanation to such a low tram-gauche barrier is a favorable arrangement with C, carbon in gauche position, enabling the formation of intramolecular hydrogen bond between a hydrogen attached to C, and the carbonyl oxygen. The distance between the oxygen and C, in a gauche conformation is 2.8 A, which is a reasonable value for a hydrogen bond. Since no hydrogens are explicitly included in the model (united atoms), this hy- pothesis cannot be confirmed. The remaining distribution functions show a similar, but more complicated structure. The tram conformation is the most probable, but the distributions have extensions towards shorter distances. These “tails” reflect the different conformations of the initial part of the chain. For example, the methyl group, n = 6, can approach as close as 5 A from the carbonyl. This, however, requires several gauche conformations along the chain, and therefore is very unlikely.

3. Dihedral angles

Distributions of dihedral angles are shown in Fig. 6. For an alkyl chain consisting of n carbons, n-3 dihedral angles are defined. Therefore within a hexyl chain there are three dihedral angles, denoted here A, B, C, where A is C, -C, -C, -C, , B corresponds to C,-C, -C,-C, , and C is C, -C,-C, -C, . It can be noted that the distributions of the three angles in the aliphatic chain show a sharp maximum at 180” corresponding to a trans conformation. The distribu-

1.0 -

0.5 -

Ii

-A

------’ B I -------- c

o.o!-++rr”;.... i 0 60 120 180 240 300 360

degrees

FIG. 6. Distributions of the torsional angles. A: C,-C,-C,-C,; B: C,-C,-C,-C,, C: C3-C,-C,-C,. Arbitrary units.

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tions of the dihedral angles are related to the radial distribution functions in Fig. 5. Therefore, the broader distribution and a relatively smaller maximum in the dihedral denoted A, is also reflected in the short distance tail in the distribution for n = 3 in Fig. 5. The relative population of tram and gauche conformations can be calculated from dihedral angles distributions. It was found that the trans populations are 0.79,0.92, and 0.95 for the A, B, and C dihedrals, respectively. In a Monte Carlo simulation of pure hexane at 25 “C, using essentially the same potential, the trans populations of the two different dihedral angles were 0.69 and 0.76.39

4. Order parame ters

The average orientation of molecules in ordered systems, such as liquid crystals, is usually defined by an order parameter. In these systems the order of the side chains is conveniently determined from the quadrupolar splittings, using deuterium NMR. 54*55 The present study is performed in the isotropic phase (no macroscopic order) and the order parameter is therefore zero. However, we need to quantify the orientational order of the different segments in the alkyl chain relative to the rigid, aromatic core. From carbon-13 relaxation measurements, which will be discussed in the next section, a generalized order parameter45 is evaluated. This parameter gives a degree of spatial restriction of the internal motion, without any assumptions about the dynamical model. From the MD simulation the order parameter tensor, S, analogous to this in the anisotropic systems, can be evaluated using the following relation:

S, = 3/2(cos Oi cos @Ii) - l/2& (7) where the brackets imply averaging over time and molecules and Oi is the angle between the ith axis of the CH, unit and the main axis of a chain. This axis is arbitrarily defined for every chain as a vector between the ether linked oxygen, 0, , and the first carbon in the chain, C, . The local, auxiliary set of orthogonal coordinates is used,22 and the axes for the nth CH, unit are defined:

z: vector from C, _ , to C, + , ; y: vector perpendicular to z, and in the plane through Cn-,,Cn,andC,+,; X: vector, perpendicular to z and y. The sum of the diagonal elements in the order parameter

tensor is zero. In case of isotropic orientation of the segment, the order parameter is zero, and it increases with increased order along the chain. For a C-H bond in a CH, unit, the order parameter is given by the relation,22 Sc, = 2/3 S, + l/3 S,,,,. In Fig. 7, order parameters, Sc,, calculated from the MD simulation, and the generalized order parameters obtained from the relaxation data are displayed for the various alkyl segments. A qualitative agreement exists, and reflects in general a degree of internal motion of the aliphatic chains in the molecule. An essentially constant value of about 0.2 for the order parameter for carbons corresponding to C,-C!, in aliphatic chains has been observed previously for various, ordered54’56 and isotropic systems.3’V33 In particular, two carbon-l 3 relaxation studies, reported recently, of the internal motions in tri-n-octylamine in isotropic phase,57 and micellar solution of tributyldodecylammonium

1 2 3 4 5 Carbon

FIG. 7. The order parameters as a function of the chain position in BHA7.

bromide,32 showed very similar decrease of the order parameters for the five first carbons, namely from 0.6 to 0.3.

B. Dynamic properties 1. Translational motion

In the MD simulation the translational diffusion coeffi- cient of a molecule was obtained from the mean square displacement according to

D=;i”, (1/6t)([R(t) -R(0)J2), (8)

where R(t) is the position of the molecule at time t. The diffusion constant for BHA7 molecule, obtained from the simulation carried out at 95 “C was 2.8 10-i’ m2 s- ‘. The experimental values were obtained from the Fourier transform pulsed-gradient spin-echo (FT-PGSE) measurements which were performed at 95 and 105 “C. The diffusion constants at these two temperatures were 3.2 lo-” and 4.5 lo-” m2 s-i, respectively. Since the length of the simulation was 100 ps,Jhe average molecular displacement during this time is 1.3 A. Such a small displacement does not une- quivocally determine the macroscopic, isotropic diffusion constant.58 Therefore, the agreement between the experimental and simulated values at 95 “C should be regarded as somewhat fortious. The values can also be compared with previously reported diffusion constants of hexa- ( pentyloxy ) triphenylene (THE5) and hexa( octyloxy ) triphenylene (THE8), diskotic molecule with masses that are, respectively, 15% smaller and 20% larger compared to BHA7. The diffusion constants for these molecules at 95 “C are: 1.5 lo-I’m* ~-‘forTHE5~~ and3.3 lo-I’m2 s-‘forTHE8.@’ This, considering the molecular masses, unexpected difference can be explained by the fact that whereas the diffusion constant of THE8 is measured in the isotropic phase, this of THE5 is reported as a “liquidlike” component in a biphasic region.

J. Chem. Phys., Vol. 96, No. 3,i February 1992


Arnold Maliniak: Discotics in the isotropic phase

2. Molecular reorientations

a. Nuclear spin relaxation. We now turn to the discussion on experimental data consisting of carbon-13 nuclear spin relaxation times and NOE factors, together with the dynamic processes that are relevant for the interpretation of these data.

All the carbon-13 signals (six aliphatic, one aromatic, and one carboxylic) were well resolved and the chemical shift assignments of the alkyl groups were made using previously reported values for heptanoic acid,61*62 based on em- pirical rules for prediction of carbon- 13 chemical shifts. The values of the chemical shifts are in good agreement with a recently reported investigation of BHA7,16 however, the as- signment of the alkyl groups clearly differs. In Fig. 8 the experimental relaxation times for carbons in the alkyl chains, measured at three different magnetic fields, are displayed. Figure 9 shows the corresponding NOE factors from the experiments at two fields. It can be noted that the relaxation time of the fifth methylene group, C, , is almost a factor of 10 longer compared with the carbon closest to the aromatic core, C, , indicating significant internal motion in the side chain. It seems, therefore, necessary to explain the dynamic processes by using a dynamic model where an overall as well as a local motion are included. The experimental data were fitted simultaneously to the expressions in Eqs. (4) and (5) using a nonlinear least squares fitting program based on su- broutine STEPIT.~~ The fitting procedure included a Monte Carlo scan of the relevant intervals for the fitted parameters, followed by an error sum minimization. All the experimental data (relaxation times and NOE factors) were given the same statistical weight in the fitting procedure. In a recent paper, 6( the problems with cross-correlation effects have been discussed, however, since the magnetization recovery curves were very easily fitted to exponential functions, we assume that these effects are not relevant in the present investigation. In the same paper,@ the relative importance of relaxation times and NOE factors in determining the parameters in the dynamic model with internal motion have been

05 I 1 2 3 4 5

Carbon

FIG. 8. Experimental and fitted carbon-13 relaxation times at three magnetic fields. The lines are included for clarity.

1 2 3 4 5 Carbon

FIG. 9. Experimental and fitted NOE factors at two magnetic fields.

investigated. The fitting analysis was therefore performed on the relaxation data only, with the results that were remark- ably identical with the fitting of all the experimental data, and as expected, the error sum in the relaxation fit considerably smaller. Since the number of parameters in this test was equal to the number of data (three parameters and three magnetic fields) such a procedure is not a fit, but a numerical test of the parameters. The carbon-proton bond length used in the expression for the relaxation rate, Eq. (4a), was 1.09 A, which is a standard value for sp3-hybridized carbon. The effect of different choice of the bond length was investigated by using 1.12 A which was obtained from the molecular mechanics (MM2) calculations. This choice did, however, not significantly affect the final values of the fitted parameters. The effect of the C-H bond length on the evaluation of dynamical parameters has been previously investigated for sur- factant molecules.33~6”

The assumption about isotropic overall motion implies that the expression for the spectral density, Eq. (5 ), contains three parameters: reorrelation time for the fast motion, r,-correlation time for the overall motion, and S-the generalized order parameter. The results from an 11 parameters fitting procedure are shown in Figs. 7 and 10, for S and rs, respectively, and the correlation time for the overall motion obtained from the fitting was 7.7 + 0.6 ns. The order parameters in the aliphatic chains have already been discussed in the previous section. The fast motion characterized by the correlation time, rY, and shown in Fig. 10, exhibits a mono- tonous behavior, clearly indicating an increasing mobility along the chain. The values indicate also that the local motion is definitely in the extreme region, whereas the overall motion is outside this region, clearly the time scale separation for the motions is fulfilled. The relaxation times and NOE factors, calculated using the parameters obtained from the fitting procedure, are included in Figs. 8 and 9. The agreement between the experimental and calculated values seems to be reasonable. Based on the experimental errors and the errors in the fitting procedure, the accuracy of the


I 2 3 4 5

Carbon

FIG. 10. The fast correlation time, T,, as a function of chain position.

parameters is estimated at 5%-10%. A check of the consistency of the model for this data set

was done by assuming that the Stokes-Einstein relation is valid and the overall motion of BHA7 can be evaluated by combining the expressions for translational and rotational motions

D=kT. 87rv? 6qr ’ rs=6kTg (9)

where k is Boltzmann constant, r7 is the viscosity, and D is the translational diffusion constant. The translation and rotational radii were assumed to be equal, and the effective radius of the molecule, r, was chosen at 10.7 A, which is somewhat smaller than obtained from the lattice constant in the liquid crystalline phase ( 11.1 A).” It was assumed that the chains due to a higher degree of motional freedom are less extended in the isotropic liquid compared to the mesophase. The overall correlation time, r,, obtained from Eq. (9) is 8.0 ns, which seems to be in reasonable agreement with the fitted value of 7.7 ns. We have investigated the effect of the slow motion correlation time, r5, on the values of the local parameters, rr and S’, by varying r between 8.5 and 12 A, which corresponds to a range of r1 of 5-10 ns. The local parameters remained virtually unchanged, with the excep- tion of the rf values for the first carbon, C,, that varied between 36 and 47 ps.

Before closing this section a final remark on anisotropic overall motion of the BHA7 will be done, this issue will also be discussed in the following section. The expression for the spectral density function for anisotropic overall motion is given in Eq. (5)) with the correlation times 7-i and r2 and A, the anisotropy parameter. The general assumption made in the model free approach45 is that the motions are statistical- ly independent, and therefore the parameters r1 , r2, and A do not have a clear physical meaning, and can merely be seen as generalized parameters for the overall motion. However, in the case of BHA7, the requirement for the time scale separation is clearly fulfilled and the overall motions characterized by the correlation times 7,) r2 can be interpreted as two


components: parallel and perpendicular to the molecular symmetry axis. The anisotropy parameter, A, can be determined from the geometrical considerations, in analogy to Woessner model for rigid molecules.66 We have performed a fitting of the experimental data with 7, , r2, and A as adjusta- ble parameters, in addition to Sand rP The correlation times obtained from this fitting were r1 = 7.4 ns and r, = 1.0 ns with A = 0.86. Since the experimental data consist of three relaxation times and two NOE factors and the number of parameters is also five, the procedure is strictly speaking not a fitting, but merely a numerical solution. The first impres- sion is that the large anisotropy seems physically unjustified, given the fact that the ratios between the elements of the moment of inertia tensor, calculated for the equilibrium conformation of the molecule, are 1: 1.8: 2.2. Note, that the anisotropy ratio reported for benzene is 1. 5-2.06’@ and the corresponding ratios for the moment of inertia are 1: 1:2. A possible explanation to this large motional anisotropy is an assumption-also confirmed in the MD simulation-about preferential pair orientation of BHA7 molecules, with the aromatic rings being parallel. Such an orientation clearly restricts the reorientation of the symmetry axis, whereas the motion about this axis is relatively unaffected. The errors in the fitting procedure were not significantly smaller compared with the isotropic assumption and therefore the anisotropic motion can not be confirmed based on the present experimental data set. The fitted parameters for the internal motion were essentially equal to the isotropic case, except for rf in the first carbon, C, , that was smaller (26 ps). An at- tempt was also made to perform a fitting procedure with fixed anisotropy rates, r, /r2, but such fits produced very large errors. b. Time correlation functions. The time correlation functions defined by Eq. (2), provide a lot of information about the molecular reorientations. These functions can be calculated from the trajectories obtained in an MD simulation, for arbitrary unit vectors fixed in the molecule, for which dynamical information is desired. In Fig. 11 a number of normalized time correlation functions is displayed, these illustrate both overall motion of the molecule and the segmental dynamics in the side chains. The unit vectors describing the overall, slow reorientations are fixed in the aromatic core are defined by:

Z: vector normal to the aromatic plane, defined by three aromatic carbon atoms; X: vector between two opposite carbons in the aromatic ring; Y: vector perpendicular to 2 and X. The local, coordinate system for the CH, units has been

defined in the order parameter section and the correlation functionsinFigs. 11(a), ll(b),and ll(c)refertox,y,andz components in this coordinate system. The correlation functions are calculated for 30 ps and it can be noted that only a short time behavior of the dynamic processes is covered. The overall motion is as expected much slower compared to the different segments of the alkyl chain. The overall reorientations are somewhat anisotropic, with the Zcomponent reor- ienting slower than the others. Note that the C, symmetry, typical for benzene, is reduced to C, when the flexible chains

J. Chem. Phys., Vol. 96, No. 3,1 February 1992



0.7 4 I 0 10 t (PSI 2 O 30

V., . 0 1'0

t (PS) 2 O 3’0

10 t (PSI

2 O 30

are attached and assumed to orientate alternatively up and down relative to the aromatic core. For a molecule with C, symmetry the Xand Y components are expected to be equal, and therefore the deviation from equality is a measure of the statistical significance in the simulation. The decays of the correlation functions for the different segments also show that the internal motions increase along the chain. There is a clear difference between the local motions of the first carbon, C, , and the rest in the chain, which has also been observed in the analysis of the experimental data, Fig. 10. Note that the x and y components [Figs. 11 (a) and 11 (b) ] of the reorientations are identical, whereas z [Fig. 11 (c) ] decays indicate a much slower motion. The reason for such an anisotropy is that truns-gauche isomerization is the most important dynamical process in the aliphatic chains, this process affects clearly the x and y more than the z component. Note that the anisotropy of the local motion can not be obtained from relaxation data assuming the two step model, since this motion is described by one single parameter, rr

The correlation functions for the overall motion exhibit more statistical noise, compared with the functions for the local motions, this is due to the fact that the latter are averaged over 192 particles (6 chains) compared with only 32 molecules for the former. The quantitative analysis of the molecular reorientations is done by defining a correlation time of a corresponding correlation function. For the overall motion an exponential decay of the correlation function is assumed and the last 10 ps were fitted to an exponential function. The correlation times obtained by this fitting were 0.75,0.90, and 1.30 ns for X, Y, and Z components, respectively. The corresponding value obtained from NMR measurements is 7.740 ns. The most serious source of the error in the dynamical parameters evaluated from the simulation is the fact that the simulation was performed at constant temperature, which implies resealing of the velocities at every time step.

In the model used for the evaluation of the relaxation data a fast motion of a segment in the side chain was superimposed on a slow, overall reorientation of the molecule. Assuming this model, the correlation functions for the reorientations of each alkyl group could be described by a sum of two exponential functions. The correlation function of this type consists of two parts, firstly a rapid initial decay due to the fast motion, followed by a second decay, (nanosecond range) corresponding to the overall motion of the molecule. The correlation functions shown in Fig. 11 re6resent only the initial part of the decay and we were, therefore, not able

TABLE IV. The correlation times for the different alkyl segments of the aliphatic chains of BHA7. Units: ps.

Site

C, G

TX TY 7,

191 189 402 118 112 286

FIG. 11. The normalized reorientational correlation functions: (a) x camp, (b) y camp, (c) z camp. The coordinates refer to the local coordinate sys- terns on each CH, group. The overall motions are included in (a). The numbers in (a) refer to the position in the alkyl chain.

C3 117 108 268 C4 114 107 239 C, 109 104 220



to fit them to a multiexponential form. Instead an effective correlation time for the local motion is evaluated by fitting the last part of the decay to one exponential function. The correlation times for the different alkyl groups are collected in Table IV. It can be noted that these decrease, as expected, along the chain, however the absolute values are significantly higher compared with correlation times for the fast motion, evaluated from the experimental data. Thex andy components are essentially identical, whereas the motion in the z direction is considerably slower.

V. CONCLUDING REMARKS

We do not believe that the simplification of the interaction model, introduced by using the united atoms instead of all the hydrogens explicitly, is significant for the results. There are also difficulties when investigating dynamic processes occurring on a nanosecond time scale, using data from a simulation performed during 100 ps. Nevertheless, the molecular dynamics simulation provided in addition to the structural data, a good qualitative picture of the dynamical processes taking place in the liquid. It would therefore be interesting to study dynamics of BHA7 in liquid crystalline phase, both by nuclear spin relaxation, preferably of a deu- terated compound, and by molecular dynamics simulation.

The combination of molecular dynamics simulations and nuclear spin relaxation was used to investigate the dynamical processes and structural parameters relevant for benzene-hexa-n-heptanoate (BHA7) in the isotropic phase.

ACKNOWLEDGMENTS

The results of the MD simulation show that the aliphatic chains in the molecule are alternatively situated up and down relative to the aromatic core, and the molecules are associated in pairs. The intramolecular radial distribution functions indicate large flexibility of the chains, which was also confirmed by the order parameters. The translation diffusion constants were measured by using FT-PGSE technique and the values were in good agreement with the diffusion coefficients obtained from the MD simulation. The diffusion constants were also compared with previously reported values for other discotic molecules of similar size.

I would like to thank Zeev Luz for valuable suggestions and discussions, Aatto Laaksonen for all the help with McMoldyn, Ulf Henriksson for many valuable comments on the manuscript, Per-Olof Eriksson for measuring the diffusion coefficients, Herbert Zimmermann for providing me with the compound, and Annika Riiffel for technical assis- tance. This work was partly supported by Carl Trygger Stif- telse fiir Vetenskaplig Forskning.

The two-step approach has been used to analyze the carbon- 13 nuclear spin relaxation data and Overhauser factors, performed at three magnetic fields, where models for isotropic and anisotropic overall motions were investigated. The conclusion is that the overall motion of the molecule seems to be adequately described assuming isotropic motion of the rigid, aromatic core. The agreement between the fit and the experimental data is not improved by introducing an anisotropy in the reorientations of the aromatic core. On the other hand, the MD simulation indicates a somewhat anisotropic overall motion. The internal motion of the segments in the aliphatic chains was described by a correlation time, TV, whereas the generalized order parameter, S, was related to the spatial restrictions of the internal motion. These two parameters, defined for each carbon in the side chain, were fitted to the experimental data. Both rf and S decrease along the chain, from the aromatic core, indicating that the relative order decreases and the mobility increases for the alkyl groups.

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