4.3 Results
4.3.1 Cylinder-forming BCPs Confined within Neutral Surfaces
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to reduce the occurrence of metastable states, we use the PT method introduced earlier. The
∆χb in the PT is chosen so that the swap probability is not too small during the simulation.
As a result, we chose smaller ∆χb for the thicker or wider simulation box.
Because we are fixing the cylinder number, the lateral size of the simulation box Lxmust be adjusted to satisfy the commensurability. We do this by choosing the natural spacing of SCFT with the corresponding effective χN. [59] For the location of the ODT points, the standard method is to calculate the order parameter according to equation (4.26). In this research, we are interested in the ODT of thin films, and thus we slightly modify this definition to perform the Fourier transform in the 2D plane at a given height. This heighthis chosen as the location of cylinders, and the mean value of W−(x, y) at height h is subtracted prior to performing the Fourier transform.
Figure 4-6 shows the L-FTS results with ¯N = 2×108, where the fluctuation is supposed to be very small. We expect the result to be very close to that of SCFT, and the ratio χ/χb is calculated to be 0.99924 using the RPA renormalization. Each simulation is performed using PT fromχbN = 13.5 to 15.5 with a uniform interval ∆χbN = 0.1. Figures4-6a and4-6b show spheres and cylinders confined in a monolayer film, respectively, and Figures 4-6c and 4-6d are the corresponding bilayer morphologies, respectively.
Each phase is clearly distinctive in Figure 4-6e. For the monolayer case, when χN is less than 13.8, it is in the disordered phase. AsχN increase, spherical phase appears in the interval 13.8∼14.4, and it goes into the cylindrical phase atχN = 14.4 or above. The story is almost the same for the bilayer film, except that the (χN)ODTis 13.8, and the transition to the cylindrical phase occurs at χN = 14.5. In the phase diagram obtained from SCFT calculation, diblock copolymers with f = 0.3 undergo phase transition from disordered phase to cylindrical phase through spherical phase, [25]. This story is confirmed by our FTS of thin films.
This result shows that the (χN)ODTdecreases as the film thickness decreases at very large ¯N value. Considering that the film is confined by neutral surfaces, our intuition suggests that the ordering is suppressed for the thinner film, and thus this result is somewhat counterintuitive. The same trend is already reported in the previous paper performing SCFT, [49] and we speculate that this ODT change is due to the enrichment of the minority phase near the interface.
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Figure 4-6: L-FTS results of cylinder-forming BCPs (f = 0.3) confined within neutral surfaces with ¯N = 2×108. (a) Spherical phase atχN = 14.29 and (b) cylindrical phase at χN = 14.79 in the monolayer film. (c) Spherical phase atχN = 14.29 and (d) cylindrical phase atχN = 14.79 in the bilayer film. (e) Plot of order parameters as functions ofχN.
Figure4-7shows the L-FTS results with ¯N = 2×106 whose fluctuation is much higher than the above case. However, we expect that the result is still not so different from that of SCFT, and the ratioχ/χb is calculated to be 0.9924 using the RPA renormalization. This simulation is also performed using PT fromχbN = 13.5 to 15.5 with a uniform interval ∆χbN = 0.1. Figures 4-7a and4-7b show spheres and cylinders confined in a monolayer film, respectively. For thicker films, they form bilayers as shown in Figures4-7c and4-7d. When compared with the previous
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results, cylinders and spheres are slightly distorted due to the increased level of fluctuation.
The order parameter plot, Figure4-7e, shows that the (χN)ODT of the two film are identical to 14.0. From this data, we can see that as the level of fluctuation increases, (χN)ODT increases, and this effect is more pronounced for the thinner films.
Figure 4-7: L-FTS results of cylinder-forming BCPs (f = 0.3) confined within neutral surfaces with ¯N = 2×106. (a) Spherical phase atχN = 14.69 and (b) cylindrical phase at χN = 15.18 in the monolayer film. (c) Spherical phase atχN = 14.69 and (d) cylindrical phase atχN = 15.18 in the bilayer film. (e) Plot of order parameters as functions ofχN.
We then perform L-FTS with a much higher fluctuation level, ¯N = 2×104, and the result is shown in Figure 4-8. For the monolayer film, the simulation is performed using PT from χ N =18.20 to 19.25. As mentioned earlier, ∆χ N between PT replica is reduced to 0.07 to
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minimize the metastability. For the case of bilayer film, it was simulated using PT from χbN
=18.10 to 18.85, and the ∆χbN is reduced even further to 0.05, because thicker film is more difficult to escape from a metastable state. The ratio χ/χb is calculated to be 0.924 using the RPA renormalization, which means that the value of the effectiveχN is from 16.7 to 17.6.
Figure 4-8: L-FTS results of cylinder-forming BCPs (f = 0.3) confined within neutral surfaces with ¯N = 2×104. (a) Disordered phase atχN = 17.07 and (b) cylindrical phase atχN = 17.52 in the monolayer film. (c) Disordered phase atχN = 16.77 and (d) cylindrical phase at χN = 17.23 in the bilayer film. (e) Plot of order parameters as functions ofχN.
One noticeable difference from the previous cases are that at this level of fluctuation, the BCPs undergo a direct phase transition from disordered phase to cylindrical phase, which is a commonly observed phenomenon in experiments where the polymers are not long enough.
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[2] Figures4-8a and 4-8b show disordered and cylindrical phases confined in a monolayer film, respectively. For thicker films, they form bilayers as shown in Figures 4-8c and 4-8d. Even when the systems are in the disorder phase (Figures 4-8a and 4-8c), we can observe localized structures with partial cylinders, but the order is not strong enough to label them as ordered phases according to the order parameter plot in Figure 4-8e. It also shows that (χN)ODTs are approximatively 17.0 and 17.3 for monolayer and bilayer films, respectively, which are much higher than the bulk ODT even after the renormalization. This increase is more pronounced for the monolayer film, and thus the monolayer goes into the ordered phase at a relatively higher (χN) value.
Regarding the ordered phases (Figures 4-8b and 4-8d), cylinders are highly distorted due to the large fluctuation compared to the case of ¯N = 2×108 or ¯N = 2×106. For a better understanding of the phase transition, we analyzed two-dimensional cross sections of each film.
Figures4-9a∼4-9d show the cross sections of the monolayer film. With these figures, it is now more visible that the system is near the disordered phase when (χN) . 17.20 and the long- range order of cylinders is more pronounced when (χN)&17.33. Along with the order parameter plot, this is another reason we identify that (χN)ODT'17.3. Figures4-9e∼4-9h show the cross sections of the upper layer of the bilayer film. Again, the long range order of cylinders becomes more pronounced for the latter two figures, and we identify (χN)ODT ' 17.0 for the bilayer case.
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Lastly, we further perform L-FTS with ¯N = 8×103 and the result is shown in Figure4-11.
This invariant polymerization index is close to the actual parameter which our experimental collaborators use, but unfortunately we are now very close to the computational limit of our FTS.
For the monolayer film, the simulation is performed using PT with uniform interval ∆χbN = 0.07 fromχbN = 20.9 to 22.23. For the case of the bilayer film, the interval of PT is ∆χbN = 0.05 and χbN changes from 20.7 to 21.65. At this ¯N value, the ratio χ/χb is determine to be 0.880 using the RPA renormalization, which suggests that the value of the effectiveχN is from 18.2 to 19.6.
In order to find the film thickness satisfying commensurability, we run multiple L-FTS with slightly different Lz size to find L∗z. In addition, we run simulations using disordered initial condition to remove the possibility that metastability affects the observed (χN)ODT change.
Figure4-10shows that the uncertainty of (χN)ODT due to the hysteresis effect is within 0.1 for the cylinder-forming polymers with ¯N = 8×103.
18.5 19 19.5
0 2 4 6 8 10 12 14 16
χN
Ψ
Initial Fields : Disordered Phase Initial Fields : Cylindrical Phase
Figure 4-10: Hysteresis effect of cylinder-forming BCPs confined within neutral surfaces with N¯ = 8×103. Order parameters are obtained from L-FTS with initial configurations of (blue) disordered and (red) cylindrical phases.
Figures4-11a and4-11b show disordered and cylindrical phases confined in a monolayer film, respectively. For thicker films, they form bilayers as shown in Figures 4-11c and 4-11d. As the effect of fluctuation increases, the change of order parameter becomes less pronounced as shown in Figure 4-11e. From this data, we temporarily locate (χN)ODTs at around 18.3 and 18.9 for monolayer and bilayer films, respectively. The trend we observed in the previous simulation is
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again confirmed. The (χN)ODT increases as ¯N decreases, and the increase is more pronounced for the case of the monolayer compared to the case of bilayer at the experimentally relevant ¯N value.
Figure 4-11: L-FTS results of cylinder-forming BCPs (f = 0.3) confined within neutral surfaces with ¯N = 8×103. (a) Disordered phase atχN = 18.70 and (b) cylindrical phase atχN = 19.13 in the monolayer film. (c) Disordered phase atχN = 18.30 and (d) cylindrical phase at χN = 18.52 in the bilayer film. (e) Plot of order parameters as functions ofχN.