a. Percolating cluster analysis

To analyze the connectivity of the morphology we use the local densities on the collocation grid, calculated from the instantaneous particle positions according toEquation 2.28.

Each grid cell that collocates more particles of the minority species than from the majority block is colored white and all others are black. Two grid cells are counted as connected if they are of the same color and touch at one of the cubes’ faces. During this analysis, periodic boundary conditions are not appliedi.e.no cells are connected via the boundaries. The resulting clusters are analyzed by their size and percolation. A cluster percolates if both of the opposing faces of the simulation box contain grid cells of the

4.1. Nonequilibrium simulations on engineering scales [156]

Fig. 4.5 Finite-size ef-fect for f = 8/32.

The graphs plot the cluster fraction, f_c, of the percolating

a circle. 0.00 0.02 0.04 0.06 0.08 0.10

Re0/^L

same cluster. A three-dimensional percolating network is established if this condition is fulfilled in all three Cartesian directions. This is sufficient as the percolating structures usually cover the full space. We do not require the stronger condition that a continuous path from any starting point in a cluster can loop back to this point via the periodic boundary conditions.

Figure 4.4graphically presents the cluster sizes for the small simulation cell,L= 10R_e, and the big system, L= 100Re. The figure shows the distribution of cluster sizes for different copolymer compositions,f. The size of a cluster is quantified by the volume it is filling (compared to the total system size) – the cluster fractionf_c. Both, clusters of the minority and majority component of the diblock copolymer are considered in this figure.

First, for reference, we discuss the cluster sizes of the equilibrium phases. Figure 4.4 includes the result for the case that the big system size,L= 100R_e, were in a defect-free lamellar configuration,f = 1/2, a perfect hexagonal phase withf = 9/32, and a gyroid morphology withf = 10/32. The lamellar configuration covers the simulation box with equally sized, nonconnected slabs. Thus, the cluster fraction of each slab is relatively small,L^∗/(2L), whereL^∗ = 1.744R_eo denotes the period of the lamellar phase and shows up on the left-hand side of the distribution graph. In the case of the hexagonal phase, the cylinders are even smaller compared to the total size, but the box contains many cylinders. The majority component in the hexagonal phase, in turn, has a significant cluster fractionfc= 1−f = 23/32 and forms a macroscopic, spanning cluster. In the gyroid phase, the minority phase forms two unconnected clusters,f_c=f /2, that span the entire volume, whereas the domain of the majority species forms a single cluster with the cluster fraction equal to the volume fraction of that species,fc= 1−f.

The nonequilibrium structures exhibit a different cluster-size behavior. On the left-hand side of the graph, the statistics of very small clusters are accumulated; they are numerous. More interestingly, for each configuration with compositions between f = 7/32 and 25/32, two macroscopically large clusters occur. The larger cluster with

fc>1/2 is composed of the majority component of the copolymer, forming the matrix, i.e., almost all polymers participate in the single, system-spanning cluster of the majority component. The volume fraction,fc, of this cluster is close to the total volume fraction, f, of the diblock’s majority component, 1−f ⪅fc. Note that the cluster fraction, fc, even slightly exceeds the volume fraction of the majority species, 1−f, because of the curvature of the internal AB interfaces. This effect is the larger the more asymmetric the composition of the diblock copolymer is.

In marked contrast to the equilibrium phases, the minority component also forms a single macroscopic cluster with a cluster fraction, f_c, that remains finite as the system size increases, for volume fractions larger than f >7/32. The cluster fraction is smaller than the volume fraction of the minority component,f_c< f, because (i) there are many additional, small disconnected clusters of the minority component and (ii) the interfacial curvature. The percolation analysis of the macroscopic clusters demonstrates that all nonequilibrium morphologies with 7/32< f < 25/32 contain a three-dimensional, percolating network of the minority and the majority component. Comparing the cluster statistics of the nonequilibrium configurations to that of the equilibrium phases, only the gyroid phase possesses this beneficial property but is limited to a very narrow composition range around f = 10/32 and f = 11/32. The conclusion is that the connectivity of the minority phase in the nonequilibrium morphologies is significantly better than that of their equilibrium counterparts. Since there are multiple strategies to stabilize these nonequilibrium structures (e.g., cooling below the glass transition temperatureTg and/or cross-linking), the three-dimensional connectivity makes them attractive for applications in batteries or fuels cells.

For f = 7/32, the minority domains of the nonequilibrium structures do not percolate in any direction during the entire course of structure formation,t <6TR, and do not form a macroscopic cluster, i.e., the largest cluster of the minority component has a negligible cluster fraction, f_c≈3.78·10⁻⁵. Clusters of this size are best compared to unconnected, spherical micelles of the equilibrium BCC phase. Because of the poor transport properties, we focus on nonequilibrium structures that contain a macroscopic, spanning cluster of the minority component, i.e. f > 7/32, in the remainder of this work.

The data highlighted in the inset ofFigure 4.4 elucidate another aspect of nonequilib-rium morphologies – a significant dependence of the cluster fraction on the system size for asymmetric composition, f <10/32. In the two systems with the most asymmetric composition, the smaller system,L= 10Reo, is characterized by a significantly smaller cluster fraction,f_c, than the larger system,L= 100R_eo. Conceiving the morphology as an irregular assembly of cylinders, we can rationalize this finite-size effect by the behavior of small clusters of the minority domain that remain unconnected to the macroscopic, percolating cluster: in the equilibrium hexagonal phase, every cylinder is not connected to any other cylinder in the system; thus no macroscopic cluster is formed. In a small simulation box, it is more likely that such an unconnected cylinder appears. The larger the simulation box the longer the cylinder has to be before it connects with its periodic image. The probability of a defect increases with the length of a cylindrical domain, and such a defect may connect it to the macroscopic cluster. As a result, we observe a rather pronounced finite-size effect.

4.1. Nonequilibrium simulations on engineering scales [156]

10² 10³ 10⁴ 10⁵ 10⁶

random walk steps ^trw

10⁻³ 10⁻²

MSD/(R2 e0trw)

f= 8/32 f= 9/32 f= 10/32 f= 11/32 f= 12/32

f= 20/32 f= 21/32 f= 22/32 f= 23/32 f= 24/32

Fig. 4.6. MSD of random walkers inside percolating clusters in the nonequilibrium morphology att= 6TR after the quench from the disordered phase. To highlight deviations from free diffusion, MSD∝trw, the MSD is divided by the time,trw, that the random walkers explore the cluster. For the three most asymmetric compositions,f, we observe a subdiffusive power-law regime at early times, followed by a linear MSD, corresponding to plateau values (dashed lines) in the graph, at late times.

In Figure 4.5we quantify this finite-size effect for copolymer compositionf = 8/32 by systematically varying the system’s linear dimension, 10Reo ≤ L ≤ 100Reo. The cluster fraction increases from f_c= 0.149±0.003 in the system of sizeL= 10R_eo to the value 0.216±0.002 for L= 100R_eo. The figure demonstrates that the simulation data are compatible with a behavior of the form,fc(L)≈f_c^∞+ const/L. This ad hoc extrapolation yields the valuef_c^∞ = 0.222±0.003, i.e. the data of the small system with L = 10R_eo underestimate this transport-relevant characteristics by 33%. This investigation demonstrates that the analysis of the size and shape of the macroscopic cluster requires extreme system sizes with billions of coarse-grained particles or a careful finite-size study.