Table 3.2: Crystallographic data ofp-6P calculated in the room-temperature herringbone phase fromN P T equilibration simulations atT = 300K. The error of these values from block averaging is less than 1\%. The middle row denotes the standard deviation of these values due to thermal fluctuations. The bottom row shows the experimental results for the\beta -phase [33].
a[nm] b[nm] c[nm] \alpha [°] \beta [°] \gamma [°] \Phi [°] \Theta H[°] \rho [g/cm3] \varphi \mathrm{C} - \mathrm{C}[°]
Simulation 0.827 0.548 2.668 90.1 101.4 89.8 17.7 61.7 1.295 15.7 Standard deviation 0.016 0.013 0.03 5.5 6.0 3.3 6.0 13.7 0.02 7.9
Experiment 0.809 0.557 2.624 90 98.2 90 18 66 1.3 20
crystal and the averaged torsional angle \varphi \mathrm{C} - \mathrm{C} in the crystal is lower by 4.3° than what is known from literature [33], which in itself is only an approximation. The deviations are smaller than previous unit-cell predictions by classical force fields of biphenyl [169], p -terphenyl [170, 171], and comparable to the best results for oligothiophenes [53, 54, 59], even though the latter calculations started already with the experimental crystal structure and not with self-assembled crystals. The deviations are also comparable to the best results in the latest crystal structure prediction blind test [55] of organic molecules. Thus, the results for the unit-cell structures are indeed satisfying.
The thermal fluctuations of the lattice parameters at room-temperature are relatively small, typically less than 2\%, as indicated by their standard deviation also given in table 3.2.
Only the thermal fluctuations of the herringbone angle\theta H exceed values of about 20\% which originate from the torsional librations of the single molecules as detailed below when we discuss the T-dependence of the crystal structures.
3.3 High temperature phases
This section reports on simulations at elevated temperatures to characterize structural phase transitions of the periodic p-6P bulk crystal. The crystal is equilibrated at various discrete temperatures ranging from 520 K to 860 K roughly in 10 K intervals for 10 to 20 ns each, depending on the state of equilibration. The pressure is set to 1 bar and is controlled using a Berendsen barostat.
An appropriate and sensitive measure for phase transitions is the isothermal heat capacity, which is the change of the enthalpy with temperature (equation 2.23). Corresponding to
Figure 3.3: Characterization of thep-6P single-crystal in theN P Tensemble simulations. (a) Heat capacity as a function of temperature. Transitions are: smectic-C\rightarrow smectic-B (smC-smB), smectic-B\rightarrow smectic-A (smB-smA) where the two-dimensional herringbone vanishes and smectic-A\rightarrow nematic (smA-nem). Transitions from experiments are indicated using the relative positions of the corresponding peaks [103, 33]. (b) Nematic order parameter S(T) and herringbone order parameter \Theta (T) as function of the system temperature T. Insets: snapshots showing the herringbone structure at room-temperature (left) and in the smA phase (right). (c) Density distribution along the nematic direction. The 18° tilt angle is the cause for the low amplitude in the density wave of the smC phase (solid red line). As the tilt angle decreases, the amplitude becomes higher and the waveform sinusoidal. When the smectic plane becomes increasingly blurry due to stronger temperature fluctuations, the amplitude decreases while keeping its sinusoidal waveform. (d) Average torsional angle\varphi av =\langle \varphi \rangle of an exemplaryp-6P molecule from inside the crystal versus temperature.
The light-blue shaded area depicts the corresponding average fluctuations\Delta \varphi 2av=\Bigl\langle
\varphi 2 - \varphi 2\Bigr\rangle
. (e) Simulation snapshots of the crystalline phases ofp-6P. Reprinted with permission from [104]. Copyright 2014 American Chemical Society.
3.3. High temperature phases
differential scanning calorimetry data of p-6P herringbone systems, [172, 33] various peaks in the heat capacity indicate transitions where the overall structure undergoes a considerable change. The results are presented in figure 3.3a: The initial room-temperature phase does not change much upon heating until T = 587 K is reached. Between T = 587 K and
T = 596 K a significant characteristic change in the density distribution along the nematic
director (figure 3.3c) paired with a very subtle change of the nematic order (figure 3.3b), are indicative of a phase transition from a smectic-C conformation to a smectic-B (smC-smB) structure. This is clearly confirmed by the trajectory snapshots in figure 3.3c which primarily show that the average inclination angle between the layer normal and the long molecular axis decreases from its\beta -phase value (18°) to an average of 0°. BetweenT = 665K andT = 677K a quasi first-order structural transition occurs with a clear discontinuity in the herringbone order (figure 3.3b). Naturally, a slight decrease of the nematic order parameter at this point coincides with the newly gained rotational freedom of the individual benzene rings. The smectic planes, even though becoming progressively blurry, still remain distinguishable. To sum up, the system undergoes a transition from a smectic-B to a smectic-A state (smB-smA). From T \approx 730 K upwards the system becomes purely nematic (smA-nem). The smectic planes become indistinguishable as can be seen in the density distribution along the nematic director as presented in figure 3.3c.
As shown in figure 3.3a, the sequence of the phase transitions is consistent with available experimental data where, qualitatively, the same mesophases in the same sequential order are reported [33, 103]. The calculated transition temperatures are within tens of kelvins of the experimental reality. We should keep in mind, however, that the finite-size simulations are not properly sampling the thermodynamic limit(N \rightarrow \infty )and employ cut-offs for the long-ranged dispersion attraction, so that the optimization of exact phase transition temperatures is in general system-size and methods dependent. Such a sensitive behavior is known already for simple Lennard-Jones systems, [173, 174] where those effects can easily lead to deviations in the tens of kelvins, and has been also observed for the organic molecule sexithiophene [59].
Furthermore, deficiencies in the intramolecular potential of the single molecule and lack of electronic polarizability also translate in the phase transition temperatures being incorrect.
Hence, given the sensitivity of the exact location of phase transition to the underlying
0
Figure 3.4: Long-time self-diffusion coefficients calculated using equation 2.38. While D\mathrm{i}\mathrm{s}\mathrm{o} = (Dx+Dy+Dz)is the usual isotropic diffusion constant,D\bot andD\| are the ones perpendicular and parallel to the nematic director. Reprinted with permission from [104]. Copyright 2014 American Chemical Society.
teractions and methods, the transition temperatures in the simulation which deviate by less than 70 K from experiments (less than 12\%), are satisfactorily described for the focus of this study, but still leave some room for improvement. With these results as a reference, further superfine-tuning of the standard force field employed here may allow optimization also of the exact transition temperatures. The remarkable fact remains that the crystal structure and phase order are correctly reproduced by this classical approach.
Due to the structural changes, the average torsion angle of the p-6P molecules in the crystal and its fluctuations considerably change with varying temperature as shown in fig-ure 3.3d. At room-temperatfig-ure the molecules are squeezed together and the angle is about 20\pm 5\circ , in agreement with experimental measurements [33]. For higher T the average value and its fluctuations increase up to 38\pm 18\circ . In a single-angle trajectory, 180\circ flips of phenyl groups are observed in the smectic-A and nematic phases (not shown) in accord with ex-periments [33]. The flips express themselves in increased fluctuations of the angle as shown by the light-blue shaded area in figure 3.3d. This analysis is an example for the detailed atomic-level structural insight into the (liquid) crystal structure of COMs provided by SD computer simulations.
As previously shown, [59] MD and SD simulations also allow the investigation of dy-namic details, important to study and understand the diffusion-controlled growth kinetics of crystals. The structural change of the crystal at high temperatures, for instance, has