Simulation methods

To mitigate finite-size effects on the morphology we use a large system size for our investigation. The length of the cubic box is L = 100Reo, where Reo quantifies the chain extension. This is significantly larger than previous simulation volumes. With a typical polymer end-to-end distance, R_eo, of about 8−27nm [35, 178], depending on the molecular weight, the simulation box represents a size of 0.8−2.7µm cubed, which can be considered engineering scale. We choose the density of chains such that the number of chains with which a reference molecule interacts matches typical experimental values. Because the number of interaction partners increases with the square root of the number of monomeric repeat units we use the value√

N¯ =ρ0R³_eo/N = 128 and denote N¯ as the invariant degree of polymerization. This quantity is invariant under changing the number,N, of coarse-grained segments to represent the molecular backbone and therefore is an appropriate quantity to relate the simulation model to experiments.

N¯ determines the strength of fluctuation effects and sets the number of chains in the simulation boxn=√

N¯L³/R³_eo= 128·10⁶. With a chain discretization ofN = 32, the total number of particles in the system isnN ≈4.1·10⁹. For reference, we also consider a smaller system of box lengthL= 10R_eo and three orders of magnitude fewer particles.

Both systems are shown in Figure 4.1, highlighting the difference in their size. The comparison of both systems allows us to identify and quantify finite-size effects.

Our large-scale simulations employ a soft, coarse-grained model in conjunction with the SCMF algorithm [33] and the SOMA implementation [34] – an OpenACC/MPI-program that enables us to efficiently utilize multiple GPUs,c.f.section 3.2.

In the soft, coarse-grained model many atomistic monomeric repeat units are lumped together into a single interaction center – a coarse-grained particle. The reduction of

4.1. Nonequilibrium simulations on engineering scales [156]

Fig. 4.1 Graphical de-piction of the simula-tion box to investigate conductivity in diblock copolymer melts. The large box has a box length of L = 100Reo, the small box is shown for referenceL= 10Reo. A magnification of the small box depicts the isosurfaces inside the system. The minority phase with the volume fraction of f = 9/32 spans a percolating clus-ter.

the degrees of freedom and the softer potentials enable simulations of large systems and decouple the chain discretization,N, from the molecular weight,M_w. The discretization of the chain contour into N = 32 particles provides a reasonable representation of Gaussian chain conformations. The details of the soft, coarse-grained model and the SCMF algorithm have been previously discussed in sections 2.1.2, 2.2, and 2.3.3. [. . . ]

In accord with previous simulations [33,55] we use the valueκ₀N = 60. Within the soft, coarse-grained model we cannot reproduce experimental compressibilities that arise from the excluded volume of individual atoms. The choice of the model parameter,κ₀N, is sufficiently large to restrain density fluctuations on the length scale of a small fraction ofRe[33]. Because this is the smallest scale relevant to the model, we expect our results for the morphology to apply to experimental systems. [. . . ]

As mentioned earlier, the three main morphologies considered for polymeric electrolytes are hexagonally packed cylinders, gyroid, and lamellar phases. At fixed Flory-Huggins parameter, χN = 30, these morphologies can be obtained by varying the copolymer composition from f = ₃₂⁷ to ¹²₃₂. The smallest composition, f = ₃₂⁷, is close to the stability region of BCC-packed spheres [1].

In experiments, microphase separation is often induced via solvent evaporation [2, 3, 179, 180], for example, after spin-coating or film-casting. Initially, the copolymer material is swollen by a solvent that is good for both components and tends to dilute the repulsive interactions between the different blocks [181]. In this solvent-swollen state, the system is assumed to be in the disordered phase. As the solvent evaporates, the copolymer density increases and so does the Flory-Huggins parameter, χ. Therefore, we mimic the process of rapid solvent evaporation by an instantaneous quench from the disordered phase into the microphase-separated region of the phase diagram. This

(a)t= 0.1TR (b)t=TR (c) t= 6TR (d)t= 61TR

Fig. 4.2. Coarsening of the minority cluster as a function of timetplotted for the small system sizeL= 10Reo and diblock compositionf = 8/32. At short timescalest < TRthe minority phase cluster is rugged and smaller in diameter. The evolution slows significantly down for t > T_R.

procedure becomes appropriate in the ideal limit that the evaporation of the fraction of the solvent that brings about the change from a disordered state to a microphase-separated morphology occurs much faster than the single-chain relaxation time and that the volume change upon evaporation is negligible.

The details of the evaporation-induced ordering process depend on the process pa-rameters, such as inter alia the phase behavior of the compressible polymer-solvent system, the dependence of the segmental mobilities on the local solvent concentration, the rate of solvent evaporation, the distance from the material surface, at which the solvent evaporates, or the film thickness [182–193]. Whereas much effort has been directed toward understanding the influence of these control parameters on the kinetics of self-assembly, the ordering process is only incompletely understood by theory and often only incompletely characterized in experiments. In view of these uncertainties, we here resort to the simplest protocol in our simulation – an instantaneous quench of the interaction parameter fromχN = 0 to χN = 30 – to induce microphase separation.

Whereas this protocol is an idealization of experiments, we expect that such a crude but generic protocol captures the universal features of structure formation in a bulk material after rapid solvent evaporation. Previous work suggests that this simulation protocol indeed captures the nontrivial sequence of states in the course of self-assembly observed in experiments [194].

Typical simulation snapshots of the time evolution of the morphology are depicted in Figure 4.2 for f = 8/32. We immediately quench a system from a homogeneous state,χN = 0, to a microphase-separated stateχN = 30. Initially, this quench results in spinodal demixing [11], i.e., right after the quench, composition fluctuations are spontaneously amplified but the fastest growing wavelength is distinctly smaller than the equilibrium periodicity [44]. Around the relaxation time T_R/2, this spinodal self-assembly is completed,i.e., the composition has saturated inside the domains and the characteristic distance between the internal AB interfaces approaches its equilibrium value from below. Here and in the following, the relaxation time, T_R ≡R²_eo/D with Dbeing the self-diffusion coefficient in the disordered phase, denotes the time that a chain’s center of mass requires to diffuse a distance,Reo. In an experimental system, this relaxation time strongly depends on chemical details, such as for example, the molecular weight of the polymer or the difference between the temperature and the glass-transition

4.1. Nonequilibrium simulations on engineering scales [156]

10¹ 10⁰

qReo

10 ⁵ 10 ⁴ 10 ³ 10 ² 10 ¹ 10⁰

S(q)

t = 0.1TR, f = 8/32 t = 6.0TR, f = 8/32

t = 61.0TR, f = 8/32, L = 10Re0

t = 0.1TR, f = 12/32 t = 6.0TR, f = 12/32

t = 61.0TR, f = 12/32, L = 10Re0

Fig. 4.3. Structure factor, S(|q|, t), of two nonequilibrium network phases f = 8/32 and f = 12/32. After the initial spinodal decomposition, the configuration become trapped in a sequence of metastable states and coarsening is slow

temperature of a component. For reference, the relaxation time of Polyethylene at T = 448K is about TR = 0.035s for a molecular weight of Mw = 30 000 g/mol and T_R = 2.195sfor M_w = 120 000 g/mol [35,195].

The subsequent changes of the morphology are protracted because the system is trapped in one of the multiple metastable state [14, 173] and the formation and growth of well-ordered grains with a (nearly) defect-free interior is far beyond the timescale considered in our simulations. Simulations of the same model at the same incompatibility and f = 1/2 revealed that structure formation is extremely sluggish, and even after 800T_R no grains with an extended, well-ordered interior are obtained [14]. Thus, the morphologies studied in the following are characteristic for an extended, experimentally relevant time interval although, eventually, we expect that well-ordered grains will form and grow.

To quantify the kinetics of structure formation we investigate the structure factor S(q, t)∝ ρ₀

4V

⏐

⏐

⏐

⏐

∫ dr[ ˆϕ_A−ϕˆ_B]e^iq·r

⏐

⏐

⏐

⏐

2

. (4.1)

Because of the isotropy in our system we study the spherically averaged structure factorS(|q|, t). This quantity is easily accessible in experiments via small angle X-ray scattering (SAXS) [17,26,31]. We show two exemplary structure factors, S(|q|, t) for volume fractions f = 8/32 andf = 12/32 inFigure 4.3. At short times, in the course of spinodal decomposition, fluctuations of the composition are spontaneously amplified.

The wave vector of the fastest growing mode is larger than that of the equilibrium lamellae and gradually shifts toward smallerq.[44]. More interestingly, we see that the structure factor changes very little between the simulation time oft= 6T_R andt= 61T_R.

Fig. 4.4 Analysis of the distribution of clusters in nonequilibrium di-block copolymer mor-phologies at t = 6TR, as function of copoly-mer composition, f. Filled bars present the data for the big system, L = 100R_e, whereas the data of the small system, L = 10R_e are shown by open bars for comparison. Data for f > ¹₂ represent clusters of the major-ity component. The in-set highlights the dif-ference between small, L= 10Re, and big sys-tems,L= 100Re. Thus, the morphologies are already trapped in long-lived metastable states.

In this work, we have simulated all configurations up to 6T_R after the instantaneous quench from χN = 0 to χN = 30, and analyzed the morphological properties of the resulting configurations. Before the quench the configurations are equilibrated atχN = 0 and κ0N = 60 for oneT_R.

4.1.3. Results and discussion

An obvious condition for ion transport is the three-dimensional connectivity of the conducting block. If three-dimensional connectivity is achieved, no preferential orien-tation of the copolymer morphology with respect to the device contacts needs to be considered. This requirement appears rather challenging because it excludes perfectly ordered lamellar and hexagonal phases, for they only allow two- or one-dimensional con-nectivity, respectively. We observe, however, that this condition is met by nonequilibrium morphologies for a wide range of compositions, f.