Matthew A. Glaser, Yves Lansac, and Noel A. Clark, Dept. of Physics, University of Colorado, Boulder

Grant D. Smith, Dept. of Materials Science, University of Utah, Salt Lake City

Abstract
An emerging trend in nanotechnology is the use of self assembling systems as templates for nanoscale patterning of materials. Smectic liquid crystals (LCs) are materials that spontaneously self-organize into lamellar structures with a well-defined periodicity of ~30 angstroms. It has recently been demonstrated that a smectic LC host can impose positional and orientational order on polymer precursors (nanosegregation) [1]. These precursors are of two types: those having chemical structure similar to LCs, which tend to segregate within the smectic layers (intralamellar monomers); and those chemically dissimilar to LCs, which segregate between smectic layers (interlamellar monomers). Subsequent polymerization of nanosegregated interlamellar monomers can produce polymer networks with a ~30 angstrom modulation of monomer density. Incorporation of intralamellar monomers into the network can impart additional structural stability. A more complete understanding of the distribution of monomers in the smectic host and the role of monomer-LC interactions in determining this distribution would facilitate chemical control of the nanomorphology of the templated polymer network. For this purpose, we have recently initiated computational studies of the distribution of polymer precursors in smectic LCs. Here, we describe development and implementation of atomistic models for monomer-LC mixtures, and present results of simulations of selected systems.

Simulated Systems
We simulated mixtures of a smectic LC host, 4,n-hexyloxyphenyl-4,n'-decyloxybenzoate (HOPDOB), with a diacrylate monomer, 1,6-hexanediol diacrylate (HDDA). The chemical structures of HOPDOB and HDDA are shown in Figures 1 and 2, respectively. We chose HOPDOB as our smectic host because it is chemically quite similar to the host used in Ref. [1], and at the same time is well characterized experimentally. HDDA is one of the polymer precursors studied in Ref. [1]. That study yielded strong indirect evidence that HDDA segregates to the interlamellar region in a smectic host. In particular, introduction of monomer into a smectic host led to an increase of the smectic layer spacing (as measured by x-ray scattering) consistent with the assumption that monomers are strongly segregated in the interlamellar region, while FTIR measurements of the orientational distribution of monomers in aligned samples indicated that monomers are preferentially oriented parallel to the smectic layers, which is also consistent with the assumption of strong interlamellar segregation.

Figure 1: Molecular structure of HOPDOB.

Figure 2: Molecular structure of HDDA.

Molecular Models
Atomic-detail, fully flexible molecular models of HOPDOB and HDDA were used in our simulations. In order to minimize the computational cost of the simulations, methylene and methyl groups in HOPDOB and HDDA were treated as united atoms. However, hydrogen atoms were explicitly included in phenyl and acrylate groups in order to accurately represent the shape and charge distribution of these groups. To enable the use of a relatively large integration timestep (see below), explicit hydrogens were assigned a mass equal to that of carbon in order to remove fast intramolecular vibrations involving hydrogen atoms. This modification of the mass distribution has no effect on equilibrium properties, and has only a small effect on the long-time dynamics of the system.

The intramolecular interaction potential employed harmonic bond stretching, valence angle bending, and out-of-plane bending interactions, with torsional potentials described by a power series in the cosine of the dihedral angle. Generic force constants from the Dreiding force field [2] were used for the bond stretching and bending interactions as well as for ring torsions and out-of-plane bending interactions. Equilibrium bond lengths, bond angles, and conformational energy profiles for rotations about backbone dihedrals were obtained from ab initio calculations at the MP2/6-31G*//HF/6-31G* level, using Gaussian 94 [3].

Van der Waals intermolecular interactions were described by a Lennard-Jones (LJ) functional form, and electrostatic interactions were computed with fixed atomic partial charges. The LJ parameters were taken from the literature [4-8]. In general, these parameters were optimized to reproduce the thermophysical properties of low molecular weight organic liquids. We calculated partial charges via electrostatic potential (ESP) fits to quantum chemical results, using AM1 electron densities obtained from Gaussian 94 [3].

Methodology
We carried out molecular dynamics simulations of several model systems with periodic boundary conditions, using the reversible multiple-timestep dynamics scheme of Tuckerman et al. [9] to integrate the equations of motion, with five distinct levels of force evaluation, corresponding to bond stretching (0.417 fs), bond angle bending (0.833 fs), dihedral torsion and out-of-plane bending (1.667 fs), short-range nonbonded (5 fs), and long-range nonbonded interactions (10 fs). The long-range nonbonded interactions included the reciprocal-space part of the Ewald sum as well as a long-range correction for the van der Waals interactions. This multiple-time step method has been found to be significantly more efficient than the widely used method of constraints. The weak-coupling method of Berendsen et al. [10] was employed to maintain constant temperature and pressure. All three dimensions of the computational cell were allowed to vary independently to maintain constant pressure, but the cell was constrained to have an orthorhomic shape.

We carried out simulations of neat HOPDOB systems consisting of 90 molecules, and simulations of mixtures of 90 HOPDOB molecules and 18 HDDA molecules (about 10% monomer by volume). Simulations of neat HOPDOB were performed at 1 atm pressure and temperatures of 355 K (in the smectic A phase) and 334 K (in the smectic C phase). Simulations of HOPDOB/HDDA mixtures were carried out at 355 K and 1 atm, with two different initial conditions (described below). All runs were of at least 1 ns duration.

Results

Neat HOPDOB
Smectic A Phase
An 1060 ps simulation of HOPDOB was carried out at 355 K, starting from a smectic A-like initial condition at a mass density of 0.7 gm/cc. Molecules were initially arranged in three layers, with the molecular long axes perpendicular to the layer planes, and with liquid-like ordering of molecular centers of mass within each layer. The smectic A layering was stable for the duration of the run, and both the average layer spacing (31.6 angstroms) and mass density (1.022 gm/cc) measured over the last 500 ps of the run are in good agreement with experiment. Thus it appears that our molecular model describes the neat smectic A phase of HOPDOB reasonably well. The initial and final configurations from this simulation are shown in Figure 3. A reduced representation of the final state is shown in Figure 4. In these and following figures, color represents depth.

Figure 3: Initial and final (after 1060 ps) states of the 355 K (smectic A) simulation with smectic A initial condition. Left: initial configuration. Right: final configuration.

Figure 4: Final state (after 1060 ps) of the 355 K (smectic A) simulation with smectic A initial condition, with molecular long axes represented by lines and molecular centers of mass by spheres. Left: projection into X-Z plane. Right: projection into Y-Z plane.

To determine whether the smectic phase is thermodynamically stable, we also carried out a 3060 ps simulation of HOPDOB at 355 K, starting from a nematic-like initial condition at a mass density of 0.7 gm/cc. Molecules were initially perfectly aligned, with liquid-like ordering of molecular centers of mass. The initial and final configurations from this simulation are shown in Figure 5. A reduced representation of the final state is shown in Figure 6. Although there is some evidence of local smectic ordering, the final configuration does not exhibit well-developed smectic A ordering. Thus it is not possible to draw any firm conclusions as to the stability of the smectic A phase at this temperature, although the smectic A phase appears to be at least metastable.

Figure 5: Initial and final (after 3060 ps) states of the 355 K (smectic A) simulation with nematic initial condition. Left: initial configuration. Right: final configuration.

Figure 6: Final state (after 3060 ps) of the 355 K (smectic A) simulation with nematic initial condition, with molecular long axes represented by lines andmolecular centers of mass by spheres. Left: projection into X-Z plane. Right: projection into Y-Z plane.

Figure 7: Final state (after 1060 ps) of the 334 K (smectic C) simulation.

Figure 8: Final state (after 1060 ps) of the 334 K (smectic C) simulation, with molecular long axes represented by lines and molecular centers of mass by spheres. Left: projection into X-Z plane. Right: projection into Y-Z plane.

HOPDOB/HDDA Mixtures
Two simulations of HOPDOB/HDDA mixtures with different initial conditions were performed. In both cases, the initial arrangement of LC molecules was similar to that of the neat smectic A sample, but in one case (interlamellar initial condition), HDDA monomers were preferentially inserted between smectic layers initially, while in the other case (intralamellar initial condition), monomers were preferentially inserted within the smectic layers. The initial mass density in both cases was 0.8 gm/cc.

Figure 9 shows the initial configuration for the interlamellar simulation, with LC and monomer molecules displayed separately. Figure 10 shows the same system after 2060 ps of simulation. Clearly, the interlamellar segregation of HDDA monomers persists over this time period, and the average layer spacing for the last 500 ps of the run (33.1 angstroms) is 5% larger than that of the neat LC. This degree of layer swelling is somewhat smaller than that observed experimentally (nearly 10% for this monomer concentration), even though the monomer segregation is apparently quite strong. This discrepancy is at least in part due to the fact that the LC molecules in one (or two) of the layers are significantly tilted with respect to the layer normal direction (see Figure 11), leading to a somewhat reduced average layer spacing. The origin of this apparent monomer-induced tilt is discussed below.

The initial configuration for the intralamellar simulation is shown in Figure 12, while the state of the system after 2060 ps is shown in Figure 13. Clearly, the distribution of monomers in this simulation still differs significantly from that of the interlamellar simulation, although there is evidence that several monomers have migrated to the interlamellar region. The average layer spacing for the last 500 ps of the run is 30.1 angstroms, which is nearly 5% less than that of the neat LC system. Again, this can be attributed to the significant tilt of LC molecules in one (or two) of the smectic layers (see Figure 14).
The observation of distinct final states for the interlamellar and intralamellar simulations clearly indicates that the simulation times are still too short Based on the rate of diffusion of dye molecules in LC hosts, several ns may be required for the establishment of an equilibrium distribution of monomers in a smectic LC host. The average potential energy in the interlamellar simulation is somewhat smaller than that of the intralamellar simulation, and the average density is slightly higher, which suggests that the interlamellar state may be more thermodynamically stable. However, no firm conclusions can be drawn without a full free energy analysis. These simulations of HOPDOB/HDDA mixtures are ongoing, and we would ideally like to continue both simulations until they converge to the same final monomer distribution.

A few words are in order regarding the apparent monomer-induced tilt observed in both simulations. We believe that this is essentially an artifact arising from the initial conditions and boundary conditions employed. Initially, each layer is constrained to contain the same number (30) of LC molecules, and slow interlayer diffusion of LC molecules guarantees that this condition is maintained for the duration of the simulation. On the other hand, monomers are inserted at random irrespective of position, so some layers start out containing more monomers than others (some monomers are initially inside the smectic layers even in the interlamellar case). Thus, all layers are effectively constrained to have the same areal density of LC molecules. On the other hand, there is a strong tendency for the system to maintain a specific mass density, both globally and within each layer. Layers containing relatively few monomers can only achieve the desired mass density within an areal density constraint by thinning, which is accomplished through molecular tilt. This artificial behavior can be overcome by working with shifted periodic boundary conditions or by tilting the layers with respect to the computational cell so that there is effectively only one layer present in the system, so that an inhomogenous distribution of monomers over layers cannot occur.

Figure 9: Interlamellar initial condition for HDDA/HOPDOB mixture. Left: LC molecules. Right: monomer molecules.

Figure 10: Final state (after 2060 ps) of the interlamellar HDDA/HOPDOB mixture. Left: LC molecules. Right: monomer molecules.

Figure 11: Final state (after 2060 ps) of liquid crystal molecules in the interlamellar HDDA/HOPDOB mixture, with molecular long axes represented by lines and molecular centers of mass by spheres. Left: projection into X-Z plane. Right: projection into Y-Z plane.

Figure 12: Intralamellar initial condition for HDDA/HOPDOB mixture. Left: LC molecules. Right: monomer molecules.

Figure 13: Final state (after 2060 ps) of the intralamellar HDDA/HOPDOB mixture. Left: LC molecules. Right: monomer molecules.

Figure 14: Final state (after 2060 ps) of liquid crystal molecules in the intralamellar HDDA/HOPDOB mixture, with molecular long axes represented by lines and molecular centers of mass by spheres. Left: projection into X-Z plane. Right: projection into Y-Z plane.

Discussion
The good agreement between calculated and experimental layer spacings and densities for neat HOPDOB in the smectic A phase indicates that the force field employed for HOPDOB is reasonable. However, the fact that smectic C ordering is not observed at lower temperatures suggests that induction interactions must be taken into account before the full phase behavior of HOPDOB and related materials can be modeled.

The stability of the interlamellar configuration of the HOPDOB/HDDA system is in agreement with experiment, which indicates a strong segregation of HDDA molecules to the interlamellar regions of a smectic liquid crystal host. The failure of the HDDA molecules to completely segregate to the interlamellar regions in the intralamellar HOPDOB/HDDA system is, on the other hand, contrary to the experimental findings. This discrepancy clearly shows that much longer simulation times (on the order of several ns) are needed for the equilibrium monomer distribution to be established.

Conclusions
This work demonstrates the feasibility of simulating liquid crystal/monomer systems, which have wide-ranging applications in nanotechnology. However, this work also points out two important requirements that must be met before modeling techniques can be usefully employed in the design of self-assembling systems: (1) the need for accurate, validated atomistic force fields and (2) the need for fast computers and algorithms to handle the very large computational requirements dictated by the large system sizes and long trajectory times which must be considered.

References
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For more information on this project, contact Matt Glaser.

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