All simulations up to this point are performed with periodic boundary conditions.
However, in Section3.2.4 we describe two different models to implement a sandwich
7At the same time also a chiral pattern is not able to form in the limitP → ∞(see Section4.4.6).
4.5 Influence of walls
dicrete flip
ρloc/ρ P = 0.125
P = 2 P → ∞ planar
z/Lz
ρloc/ρ 0 2 4
0 2 4−0.5
0 0.5
−0.5 0 0.5
Figure 4.22: Comparison of the two models for the sandwich geometry. The local number density is calculated in 10 boxes along thez-axis (N = 1000, σ/Lz = 0.1).
geometry. In the following we want to study the influence of such boundaries.
We perform simulations of a small system (N = 1000, ρ= 1) and investigate the vertical density profile which measures the local density in layers parallel to the walls (Fig. 4.22). Both models show a vanishing local densities close to the walls due to the repulsion (Eq. 3.50).
The model with planar anchoring (Eq. 3.52) shows large accumulations of particles close to the walls (Fig.4.22, left panel). This effect is strongest for the deterministic model (P → ∞), followed by the isotropic phase (P = 0.125), and it is weakest in the nematic phase (P = 2). The wall-accumulation is directly induced by the anchoring mechanism: at the top wall all particles align along ˆx so that it is hardly possible for a particle to leave the vicinity of the wall again. At the bottom wall, the inter-particle alignment can lead to local alignment of all particles close to the wall which again keeps all these particles together. The wall-accumulation is strongest for the deterministic system (P → ∞) where particles are strongly trapped at the walls because no rotational noise is applied. The weakest accumulation is found for the nematic phase because the walls will induce a global alignment perpendicular to the z-axis preventing particles from approaching the walls. On the contrary in the isotropic phase, the orientations of the particles are equally distributed into all directions which means that at all times particles are approaching the walls. At the same time the particles being already close to the walls are subject to stochastic
two directions
Figure 4.23: Nonequilibrium phase diagram (analogous to Fig. 4.1) of SPPs in a sandwich geometry using the discrete flip model (γ = 0.1, 241/γF = 0.1).
rotational noise which can possibly take them away from the wall. However, these two effects do not cancel out because the particle-wall alignment mechanism traps the particles at the walls.
The second model forSPPs in a sandwich geometry is termed “dicrete flip” (Eq.3.54) and shows a homogeneous distribution of the local density (Fig. 4.22, right panel).
The resulting density profile does not change for different Péclet numbers. We hence use this model in the following to investigate the influence of walls onto aligningSPPs.
The phase diagram (Fig. 4.23) is very similar to the one of the particles in a simulation box with periodic boundary conditions (Fig. 4.1). Mainly two domains can be identified: An isotropic phase forms for low Péclet number and a nematic phase evolves where the Péclet number is high. The transition line again has a negative slope in the (ρ,P)-plane so that the critical Péclet number decreases with increasing global density. Close to the transition line (and especially at low densities) the phase coexistence and a traveling wave are found.
The main difference of the phase diagram of the system with walls compared to the periodic boundary system is the absence of the chiral simulations scattered in the nematic phase. However, out of all performed simulations, we find one in the nematic phase (P = 4.1, ρ = 1.75) which shows a considerably smaller global nematic order parameter than all surrounding simulations. Visual inspection of the corresponding snapshot (Fig.4.24b) reveals a very interesting pattern: The simulation box is subdi-vided into several distinct parts. More than half of the box is populated by a rather dense, locally nematically ordered “stream” of particles with the local director being aligned perpendicular to the walls. The rest of the box is rather empty with only a few particles which do not show nematic alignment on a larger scale except for a second dense “stream” close to the bottom wall of the simulation box oriented along the simulation box. The two local nematic directors of the two “streams” hence form
4.5 Influence of walls
dloc,z
dloc,y
dloc,x
components of ˆdloc
ρloc/ρ
(a) Vertical profiles (measured in 50 slices) of local nematic order parameter, local den-sity, and components of local nematic direc-tor.
localnematicorder parameterSloc
0 0.5 1
(b) Snapshot with local nematic order pa-rameter calculated in 203 boxes (like in Fig. 4.2)
Figure 4.24: Two-directional pattern in the system with sandwich geometry (P = 4.1, ρ= 1.75).
an angle of π/2. For further analysis the vertical profiles of local density and local nematic order are considered: The simulation domain is subdivided into 50 equally spaced, horizontal slices to calculate the local density, local nematic order, as well as the local director within each slice (see Fig. 4.24a). The vertical profiles of the components of the local nematic director substantiate what we described from the visual inspection of the snapshot: The local director almost everywhere in the box is aligned along the z-axis, i.e. the particles move perpendicular to the walls. The lower part of the simulation domain, however, shows a local alignment in the x−y plane (vanishing z component of ˆdloc) but not along one of the axes. The vertical profile of the local density is homogeneous in the upper part of the box and increases towards the lower wall. It is a gradual increase and not a sharp transition like for the components of the local director. The shape of the vertical profile of the local density reflects the fact that close to the bottom wall a second “stream” of particles is formed next to the vertical one so that the number of particles in a horizontal slice increases. The local nematic order parameter shows a non-monotonic vertical profile with a high value (Sloc ≈0.8) in the upper part of the simulation domain where the particles are mostly aligned vertically. A minimum of Sloc is reached where the local director transitions from vertical to horizontal alignment and hence the system ap-pears disordered. Close to the lower wall the nematic order parameter increases again with a maximum value of Sloc ≈ 0.5 in the vicinity of the wall where the horizontal
“stream” dominates.
Sχ
S
globalorderparameter
time ×105
0 0.5 1 1.5 2
−1
−0.5 0 0.5 1
Figure 4.25: Temporal evolution of the global order parameters in the simulation which shows two main directions of local nematic directors (sandwich geometry, see Fig. 4.24).
The temporal evolution of the nematic and chiral order parameters shows a sur-prising stability of this configuration (Fig.4.25): it lasts for at least 2×106 time steps (∆t = 0.1) which is comparable to the stability of the chiral pattern in the system without walls (Fig. 4.11a). Both order parameters fluctuate in time around interme-diate values ofS ≈0.5 and Sχ ≈ −0.3. These values reflect the partial nematic order in the system, as well as the occurrence of a chiral pattern even though the particles do not form a helix like in the case with PBCs. We find that the walls suppress the formation of chiral patterns in terms of helices but on rare occasions a chiral pattern can emerge in the nematic area of the phase diagram which is composed of differently oriented, nematically ordered domains.