b. Diffusive transport properties

To illustrate the transport properties of the nonequilibrium structures, we employ a simple random-walk model. A random walker can attempt a move in any of the major Cartesian directions with equal probability, respecting the periodic boundary conditions.

Diagonal moves on the lattice are not permitted. These random walkers mimic the ions diffusing inside the conducting cluster. To this end, we randomly place 8000 random walkers inside the percolating cluster. This number suffices to (i) sample all parts of a configuration and (ii) obtain good statistics. These walkers perform a random walk on a lattice. Unless noted otherwise, this lattice coincides with a collocation grid of the density. Any proposed move that would place the walker outside of the cluster is rejected. Each of the walkers attempts one million displacements,t_rw ≤10⁶.

0 1 2 3 4 5 6 in-creases at early times,t, as a macroscopic cluster forms. At later times the diffusion constant is almost constant or slowly increases.

It is important to differentiate between the two independent times: (i) the time, trw, that the random walkers diffusive inside the percolating clusters and that is measured in units of attempted random displacements, and (ii) the time,t, that the copolymer morphology evolved after the quench from the disordered phase. This latter time is measured in units of the relaxation time, T_R of a polymer molecule. The diffusion analysis is always performed inside a “frozen” configuration at a fixed timet.

Figure 4.6 plots the MSD of the random walk as a function of attempted, random displacements,trw. Inside the percolating cluster, the MSD linearly increases, MSD→ 6Dtrw, in the limit of long times, trw → ∞. Inside of the cluster of the majority component,i.e., the matrix, we observe this linear relation for all times.

For the diffusion inside the percolating cluster of the minority component, however, a subdiffusive behavior MSD ∝t^2/drw^w with d_w >2 is present on short timescales. Only later the MSD becomes linear. These characteristics are well-known for diffusion in overcritical percolation clusters [196]. The subdiffusive behavior is observed on length scales smaller the correlation length of the infinite cluster, whereas a linear MSD is observed on larger scales. Thus, the interpretation is that on short time and length scales the fractal geometry of the percolating cluster is relevant for the diffusive transport, whereas the percolating cluster acts like a homogeneous, three-dimensional medium on large time and length scales.

The crossover between the subdiffusive behavior and the linear increase of the MSD defines a characteristic time and a length scale. Extrapolating the two regimes with straight lines in the double-logarithmic plot (c.f. Figure 4.6), we quantify the crossover time,t^c_rw, and the concomitant length scale, l^c=^√MSD(t^c_rw). The subdiffusive regime is more extended for more asymmetric compositions,f. For the three smallestf, we obtain l^c8

32 = (18±1)R_eo,l^c9

32 = (12±1)R_eo, and l^c10

32 = (13±3)R_eo. This effect is not surprising as the cluster of the minority component becomes narrower and more rugged, the smaller the diblock composition,f, is. Importantly, the crossover length is larger than 10R_eo,i.e.the system size of the smaller configuration. Consequently, these smaller systems cannot capture the complete diffusion phenomena.

4.1. Nonequilibrium simulations on engineering scales [156]

(a)Lamellar phasef= 1/2 (b) Hexagonally packed cylindersf= 1/4

Fig. 4.8. Reference configurations to obtain the diffusivity of a lamellar and hexagonal phase.

Some particles of the majority component have been removed to highlight the structure.

To further analyze the transport properties, we compare the long-time diffusion coefficient of the random walkers inside the percolating cluster with the optimal diffusion coefficient, Dopt = ∆L²/6 =R²_eo/216 that is obtained if the clusters cover the entire volume. The ratio between the actual diffusion coefficient and the optimal one, D/D_opt, quantifies the relative conductivity of a given morphology. This method does not account for details smaller than the grid resolution ∆L=Re/6, liquid packing of the monomers and local cluster shape, but it allows for an efficient analysis of the large systems.

Effects of interface roughness on the molecular scale or specific mechanisms of the particle dynamics are ignored but a careful, previous study indicated that additional characteristics of the diffusive transport do not result in significant, qualitative deviations compared to the simple random walker model [171,197].

Figure 4.7presents the so-determined diffusion coefficient inside the nonequilibrium morphology as a function of the time,t, after the quench. The diffusion inside clusters of the majority component,i.e., the matrix, is rather independent of t and increases with the volume fraction, f, of the component forming the cluster. For clusters formed by the minority component, initially, the diffusion coefficient rapidly grows with time fort < T_R but subsequently increases significantly slower.

The reason for this characteristic is the formation of A and B domains, starting form the disordered phase. We analyzed and discussed the dynamics earlier: after the formation of the A and B domains, the system becomes trapped in a metastable state [14,173].

328 9

In the following, we compare the diffusion coefficient of the nonequilibrium structures at t = 6T_R to the values of the equilibrium phases. Two equilibrium structures are depicted in Figure 4.8. First, an external field is employed to guide the structure formation into these defect-free domains. In a second step, the obtained morphologies are equilibrated without an external field, and we adjust the box dimensions to be commensurate with the equilibrium spacing of these equilibrium mesophases. We achieve the optimal box dimensions by ensuring that the diagonal components of the bonded stress tensor are the same, which is expected for an isotropic melt [33]. The periodicity of the mesophases allows us to study smaller system sizes compared to the nonperiodic, nonequilibrium morphologies. The lamellar configuration with f = 1/2 has dimensions of Lx=Ly = 4Reo andLz = 5.232Reo with n= 10715 polymers, see Figure 4.8a. The hexagonal configuration with f = 1/4 has dimensions of L_x= 3.63R_eo,L_y = 4R_eo, and Lz = 6.28735Reo with n= 11686, see Figure 4.8b. All other parameters are identical to the nonequilibrium structures.

In the two equilibrium phases, diffusion is, of course, limited to two-dimensional sheets in the lamellar phase and one-dimensional cylinders in the hexagonal phase.

Nevertheless, the three-dimensional MSD is used for the determination of the diffusion constant D, enabling a comparison to the nonequilibrium morphologies. Neglecting interface effects, one ideally expectsD/Dopt = ²₃ and ¹₃ for the lamellar and hexagonal phases, respectively.

In order to quantitatively analyze the diffusion and compare the nonequilibrium structures to the equilibrium phases, we use a finer discretization and filter the local interface roughness as detailed in the Appendix of this section 4.1.5a. The obtained results of the simple random walk in the filtered configurations are close to this prediction, seeFigure 4.9. This behavior is in agreement with previous work by Shen et al. [171].

Figure 4.9depicts the obtained diffusion constantsD/Dopt as a function of diblock composition, f. The data show a general trend – the higher the volume fraction f of the conducting component is, the higher is the diffusivity. This effect is compatible with

4.1. Nonequilibrium simulations on engineering scales [156]

the following rationale: at large f > 1/2 the diffusion in the cluster of the majority component is slowed down in the vicinity of the interface to the minority component.

The minority component forms channel-like domains of cross sectiond², wheredis set by the molecular weight but is largely independent off. The lengthL of all channels is given by (1−f)V =Ld². In the interfaces of these channels, the diffusion is reduced from D0 to D−. Let w denote the interface width, then the corresponding interface volume scales like dLw∝(1−f)V w/d. Thus the volume-averaged diffusion constant is

D≈ D₀f +D−(1−f)w/d f+ (1−f)w/d ≈D₀

[1−(1−f)w d

(1−D−

D₀

)+. . .

] (4.2)

Such a linear relation can be observed for larger volume fractions,f, of the conducting component in the graph. For these strongly asymmetric compositions, diffusion in the cluster of the majority component is limited by the accessible space but the shape of this space is less relevant.

On the other hand, for volume fractions smaller thanf <12/32 the diffusivity drops significantly in the nonequilibrium structures. In this case, the shape and connectivity of the three-dimensional network limit diffusion, resulting in a smaller conductivity of the nonequilibrium morphology compared to the equilibrium phases, although the defect-free equilibrium phases only allow for one- or two-dimensional transport. It is the complex, tortuous geometry of the clusters of the minority component in the nonequilibrium structures that reduces the diffusivity.

A method to characterize the shape of diffusion paths inside a morphology is the tortuosity,τ. In general, the tortuosity is a measure to quantify how twisted or bent a structure is. In the context of transport properties, it can be interpreted as the deviation of transport paths from a straight line. For perfectly straight paths the tortuosity equals one and increases for bent paths. A common way to define the tortuosity is via the conductivityσ [171,174].

σ = ϕσ_opt

τ (4.3)

where σ_opt is the conductivity of the homogeneous bulk material, and ϕ denotes the accessible space of the conducting phase, i.e., the cluster fraction ϕ ≡ f_c. In the limit of ions diffusing noncooperatively and with an infinitesimally small flow rate, the conductivityσ can be related to the diffusion constantD. Thus, Equation 4.3can be rewritten in the form [171]

τ = D_opt

D . (4.4)

The inset of Figure 4.9presents the tortuosity of the nonequilibrium morphologies as a function of the copolymer composition. It confirms the prior observations: for high volume fractions the limiting factor is the accessible space whereas for low volume fractions of the conducting component the tortuosity increases and hinders free diffusion.

We also note that the tortuosity of the equilibrium phases is in both cases close to 3/2 and 3, respectively – in accord with the straight-path interpretation of the tortuosity.

Interestingly, comparing data from the small system,L= 10R_eo, and the large system,

(a)Cluster filled with liquid before it is vertically drained from the bottom.

(b) The same cluster, but the liquid has been drained from the bottom – only dead-ends re-main filled.

Fig. 4.10. Illustration of the dead-end analysis, draining a filled percolating cluster of the minority component, f = 9/32. The system size isL= 10Reo. The green interface of the cluster is cut away in the upper half of the box to reveal the dead-end cluster inside the configuration (blue).

L= 100Reo, we observe small discrepancies. For the higher volume fractions, f, of the conducting component it is sufficient to investigate the smaller system to accurately determine the diffusive properties. For the small volume fractions, however, the small system overestimates the tortuosity, in accord with the finite-size effects presented in Figure 4.4.