Once the simulations have reach equilibrium, the systems were analysed in a variety of ways. By monitoring the energy of the simulation, it was possible to tell when a system had reached equilibrium as the energy of the simulation was constant. For example, simulations started from an isotropic phase showed a sharp decrease in energy on the formation of a nematic phase whereas simulations started from a perfect crystal showed an increase in energy as the order of the system was lost on the formation of a nematic. For bulk systems, other parameters such as the nematic order parameter, ̅ , can be monitored to
investigate equilibration.
B
ULK ORDER PARAMETER1.4.1
The bulk order parameter (1.1.1) was calculated over the simulation box to classify the phase in some of the systems investigated. The order parameter was calculated for each analysis cell and also a global order parameter was calculated for bulk mesogen-only systems, ‘slab’-like systems and confined cylinders. The global bulk order parameter was not calculated in the majority of the confined systems as it is meaningless due to the spherical or toroidal geometry of the system.
The bulk order parameter was calculated by calculating the a Q-tensor over all of the mesogens in the system;
( ) ∑ ( )
(1.4.1)
62
( ) (1.4.2)
so that and the bulk order parameter is taken to be
〈 ( )〉 (1.4.3)
V
ISUALISATION1.4.2
For many of the confined geometries investigated, as stated previously, the bulk order parameter does not provide any meaningful information. For these geometries a measure of the local order is required to locate the defects. To do this, the simulation cell is split into cubic analysis cells that are set to be the same as the cubic cells used in the neighbour list. The size of the analysis cells was a balance between two main factors; smaller analysis cell size meaning the visualisation and position of the defects is of higher resolution whilst a larger analysis cell size means there are more particles in each analysis cell and so less MC cycles are need in order to gain any statistical data about the local order and direction of the director.
In order to visualise the systems simulated and locate the defects, the method proposed by Callan-Jones et al[92] was employed and then visualised using Paraview version 3.10.1. This method uses a modified order parameter tensor to calculate three Westin Metrics which are used to represent the linear, planar and spherical order[93].
In each analysis cell a Q-tensor was calculated for each analysis cell (1.4.1)and diagonalised to give a matrix in the form
(
)
63 where S is the order parameter and η is the biaxiality parameter (in a uniaxial nematic, η=0) and is equivalent to that shown in (1.4.2). In order to visualise the simulations, the tensor used must be greater than or equal to zero at all points. To achieve this, a modified tensor is used;
( ) (1.4.5)
where and From , the three Westin metrics,
and , can be calculated.
(1.4.6)
( ) (1.4.7)
(1.4.8)
The three Westin metrics must all have values between 0 and 1, and In a well ordered uniaxial nematic phase, and In an isotropic phase ( ), and so and which is equivalent to a sphere. In reality, is always slightly above zero for a small system due to the presence of some degree of orientational order in a liquid. The same is true for ̅ for an isotropic phase in a small system.In this thesis only uniaxial molecules were investigated, i.e. (as is a measure of biaxiality), and so .
D
IRECTORV
ISUALISATION1.4.3
The director was visualised using stream lines whose trajectory sweeps along the
eigenvector field corresponding to . The streamlines were started from random points in the simulation cell and were chosen so they passed through all regions where there is a non-zero density of mesogens. The director stream lines were used to detect the locations of any escaped defects with s=+1 that were not found using the method below. The stream
64 lines were also used to distinguish between defects of s= and s= by studying how the director rotates around the defects (Figure 1.1.7).
D
EFECTD
ETECTION1.4.4
The defects were defined as areas where there was density of mesogens, but is less than a threshold value. The threshold value was chosen as it showed a surface in the defect locations but not at the nematic-water interface. The threshold value used was dependent on the temperature of the system. For simulations run at
⁄ , a threshold of was used unless otherwise stated. For simulations run at a higher temperature a much lower threshold value was used as the system was much more disordered and the reverse is true for systems at a lower temperature.
Using this method to detect the defects was very successful for defects of strength , it was not always possible to detect defects of strength s= 1. The s=+1/2 defects were easier to detect due to the different nature of the s= defects and the s= 1. The half strength defects are axial disclination lines through the nematic whereas the s= 1 defects are escaped defects on the surface. The defects on the surface are point-like in two dimensions and so tend to have a smaller region of disorder than the axial s= defects.
65
T
HE INTERACTION POTENTIAL
2
In this chapter the basic features of the mesogen-mesogen interaction used in this thesis are given, as well as the mesogen-water and water-water interactions used for studying nematic shells. A simple model based on hard spheres with an embedded orientation vector allows for large system sizes of approximately 100,000 particles to be investigated. The chapter then goes on to discuss some preliminary results testing the behaviour of our proposed model with bulk systems and slab-like systems with flat interfaces. An off-lattice model allows for deformation of the spherical shell to occur. By using an off-lattice model, unlike the lattice based analogues, it is also possible to vary the elastic constant ratios.