5.2.1 Commensurate Thickness: L
z/λ = 1 and 2
Now we turn to the situation where the walls remain homogeneous but one or both of them prefer the A-component. To model an A-attractive wall we change the prefactor ˆǫw in (5.1)
5.2 Parallel Lamellae between Homogeneous A-preferential Walls
Figure 5.6: Isosurfaces of the perpendicular lamellar structure in the thin film with neutral walls at the time 20000 MCS,Lz =λ. The structure is equilibrated and has a local order which extends over two periods.
to ˆǫAw = ǫw−δw for A-monomers and ˆǫBw = ǫw+δw for B-monomers where the parameter δw
is positive so that the A-preference of the wall means that the wall has a stronger repulsion for B than for A monomers. Choosing δw = 0.5, the energetic preference of A-monomers is strong enough to overcome the essentially entropic alignment effect discussed before, that molecules near the wall have reduced orientational degrees of freedom. First we consider films of a commensurate geometry with thicknesses Lz =λ (lateral dimensions Lx = Ly = 4λ and the number of molecules M = 3331) and Lz = 2λ (Lx = Ly = 3λ, M = 3747). We made 5 independent runs for symmetric films with A-attractive upper and lower walls and 10 runs for asymmetric films with A-attractive upper and neutral lower walls.
In Fig. 5.7a the corresponding A-monomer density ρA(z, t) is plotted across the symmetric film of thickness Lz = λ. Layers of A-monomers adjacent to the walls rapidly form and get separated by a B-rich domain. Notice a peak of the A-density in the center of the film which emerges at initial times of the simulation. Its origin is due to the fact that the initial formation of the A-rich layers goes through a fast reorientation of the molecules located close to the walls
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35
ρ
A(z, t)
z
a)
4×101 MCS4×102 8×102 2×104
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 5 10 15 20 25 30 35
ρ
A(z, t)
z
b)
4×101 MCS4×102 8×102 2×104
Figure 5.7: (a) Time evolution of the A-monomer density in a film of thickness Lz = λ with A-attractive walls, for χ = 0.45 (b) same, but with A-attractive left and neutral right wall. Systems with both types of walls develops A−B−Alamellar structure across the film.
which leads to some depletion of A-monomers in the adjacent regions. Diffusion of A-blocks from the center of the film is not fast enough to compensate this depletion at initial times.
The peak decreases in hight and disappears at 800 MCS. The final profile of the A-monomer density which becomes equilibrated at about 2·104 MCS has two well defined peaks at the
5.2 Parallel Lamellae between Homogeneous A-preferential Walls
Figure 5.8: Schematic representation of the parallel lamellar structure observed between two homo-geneous A-attractive walls, the film thicknessLz=λ.
walls and decreases towards the middle of the film where it has zero values. In the middle of the film we observe a peak in the B-monomer density (not presented here), a clear indication of lamellae parallel to the walls. A schematic representation of the structure is shown in Fig. 5.8.
When the wall at z =Lz is replaced by a neutral one, the equilibrium density profile (40000 MCS) essentially remains symmetric, see Fig. 5.7b. At 400 MCS we observe a peak of the A-monomer density located near the middle of the film which then moves due to reorientation of the molecules towards the neutral wall. Thus the A-attractive wall induces layering almost as in Fig. 5.7a. Development of the A-rich layer near the neutral wall, however, takes much longer time than in Fig. 5.7a. The whole situation is reminiscent of wall-induced spinodal decomposition in films of binary polymer blends, at least in its early stages [Eudiss], [Pur97], [Fis98].
In Fig. 5.9 we plot the z dependent A-monomer density in symmetric and asymmetric films of thicknessLz = 2λ. In the symmetric film, Fig. 5.9a, we observe A-rich layers already formed at the walls at the time 40 MCS, then at the time 400 MCS two additional peaks appear, reflecting two A-rich layers in the middle of the film. The peaks move towards each other (800 MCS) and then coincide to form one layer in the film center at the final time 2·104 MCS.
The ordering process is different in the asymmetric film, Fig. 5.9b, where we have an “ordering wave” resulting in appearance of the first layer at the A-attractive wall, a gradual increase of the second layer, its movement towards the center of the film followed by a development of the third layer at the opposite neutral wall. The final equilibrated structures in both films are quite similar and consist of lamellae parallel to the walls, see Fig. 5.10, the equilibration time in the asymmetric film is two times larger as that in the symmetric film.
5.2.2 Incommensurate Thickness: L
z/λ = 1.5
So far the thickness of the film with A-attractive walls (or A-attractive lower and neutral upper walls) was an integer multiple of the bulk lamellar periodicity so that the parallel lamellae observed were not compressed or stretched. Now we study the microphase separation in films of thickness incommensurate with the bulk periodicity. In Fig. 5.11a we plot the A-monomer density as a function of z in the symmetric film of thickness Lz = 1.5λ with A-attractive upper and lower walls. The time evolution of the A-density profile runs quite similar to that of the parallel lamellar structure formation in Fig. 5.9 except that there is only one peak in the middle at all the times. The final structure is compressed parallel lamellae with the periodicity λf = 3/4λ.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 10 20 30 40 50 60 70
ρ
A(z, t)
z
a)
4×101 MCS4×102 8×102 2×104
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 10 20 30 40 50 60 70
ρ
A(z, t)
z
b)
4×101 MCS4×102 8×102 2×104 4×104
Figure 5.9: (a) Time evolution of the A-monomer density in a film of thickness Lz = 2λ with A-attractive walls, forχ= 0.45 (b) same, but with A-attractive left and neutral right wall. The systems develops A−B−A−B−A lamellar structure across the film.
The time evolution of the A-density profile in an asymmetric film of the same thickness, Fig. 5.11b, shows the same features as in the asymmetric film of the Fig. 5.9b with an ordering wave of parallel lamellae moving from the A-attractive lower wall towards the neutral upper one. The A-density profile gets equilibrated at about 2·104 MCS having one peak at the lower