## Reports: DNI756499-DNI7: Fundamental Understanding of Anisotropic Ion Transport in Charged Block Copolymer Microdomains: Large-Scale Mesoscopic Simulations of Phase Diagrams, Ion Diffusion Pathways, and Electric-Field Effects

**Shangchao Lin, PhD**, Florida State University

**Introduction and
Progress Summary for Year 1**

Ionic
conductivity in block copolyelectrolytes highly depend on the morphology of
self-assembled microdomains at the molecular level. However, the influence of
electrostatic interactions on morphology and ionic conductivity remains an
obscure feature of this type of charged block copolymer (BCPs).^{1-3} Current
theoretical understandings of phase diagrams of BCPs, whether obtained from
mesoscopic dissipative particle dynamics (DPD) simulations^{4-5} or
self-consistent field theory (SCFT)^{6-7}, haven’t
addressed the inherent difficulties of capturing the following elements
simultaneously: (i) discretized local charge distributions and electrostatic
interactions, (ii) quantifying the ionic transport process, and (iii) direct
correlation between the ionic conductivity and the morphology anisotropy.^{8-10} Therefore,
there are urgent needs to develop reliable computational tools to allow *de
novo* design of charged BCP systems.

In Year 1 of this project, we
have predicted the morphology of self-assembled block copolyelectrolytes using
a modified DPD simulation framework, considering both electrostatic
interactions and explicit ion diffusion. We identified microphase transitions
based on the directly computed structure factor patterns. Finally, we mapped
out a few phase diagrams under various experimentally-controllable parameters,
including block A ratio (*f _{A}*),
Flory-Huggins parameter for A-B repulsive energy (

*χ*), block-A charge fraction (

_{AB}N*ϕ*), and dielectric constant (

*ε*).

_{r}**Computed Phase Diagram of Neutral Diblock Copolymers**

To
validate our DPD simulation framework, we first predicted the morphology of neutral
BCP systems. Ordered microphase structures, including lamellar L, gyroid G,
cylindrical, C, spherical, S, and the corresponding inversed phases (B-rich), are
observed as shown in **Fig. 1**. These phases were identifying by viewing the
equilibrated simulation snapshots, which is further confirmed using computed
structure factors (see **Fig. 2**) that mimic experimental X-ray diffraction
spectra. We conclude that our DPD framework and the identification of phase
diagrams conform well to both experimental and theoretical SCFT results.

**Figure
1:**
DPD-predicted phase diagram (as functions of *f _{A}* and

*χ*) of a neutral A-B BCP system with representative simulation snapshots. Solid black lines show predictions using the SCFT by Schick

_{AB}N*et al.*

^{11}DPD simulation results are corrected here to account for the low-molecular-weight effect as proposed by Matsen.

^{12}Only one of the two blocks (A in red and B in blue) is shown in the snapshots for clarity.

**Figure
2:**
Structure factor patterns for neutral BCP microphases computed using DPD simulated
trajectories. The characteristic peak locations for each microphase are marked
by numbers. The *x*-axis represents the normalized *k* value by the
location of the first primary peak, *k**.

**Computed
Phase Diagram of Diblock Copolyelectrolytes**

After
validation the DPD framework for neutral BCP systems, we considered electrostatic
interactions when positive counterions are introduced into the system and block-A
beads are uniformly and negatively charged. Such block copolyelectrolyte
system can be characterized by the dielectric constant (to reflect the impact
from solvents), the ionic valence (for ions like Li^{+}, Mg^{2+}
and Al^{3+}), and the charge fraction of the block A (*ϕ*, defined
by the ratio between the number of counterions and that of block-A beads). We
found that the ionic valence is directly proportional to the charge fraction of
block A due to electroneutrality, and therefore, we only varied the charge
fraction of block A in this project. The DPD-predicted phase diagrams for the
charged BCP system are shown in **Fig. 3 **when *ϕ* = 1:4 and *ε*_{r}
= 1.1 for a water-type solvent environment in the DPD framework.

**Figure 3:** DPD-predicted phase
diagram of the charged BCP system (A* ^{n–}*B) with representative
simulation snapshots. The dashed lines are guides to the eye. Only one of the
two blocks (A in red and B in blue) is shown for clarity, while the counterions
are always shown in green.

** **

In
addition, have also looked into charged BCP systems with when *ϕ* = 1:2
and 1:8 (both under *ε*_{r} = 1.1), as well as when *ϕ*
= 1:4 but with a reduced *ε*_{r} = 0.275. The DPD-predicted
phase diagrams are shown in **Fig. 4**. In general, these phase diagrams share
two common features. First, at high *χ _{AB}N* values when

*f*

_{A}is increased, the phase transition follows the same order (D→S→C→G→L→G’→C’→S’→D) as that for neutral BCPs. Second, they all exhibit leftward (regarding boundaries between ordered phases or order-to-order transition, OOT) and upward (regarding order-to-disorder transition, ODT) shifts, compared to the neutral BCP phase diagram. When increasing or decreasing the charge fraction value

*ϕ*to 1:2 or 1:8, respectively, such shifts of the boundaries between the ordered phases become more or less significant.

Excluded
volumes contributed by the counterions which bind closely to block A
electrostatically, increase the effective *f*_{A} values in the
phase diagram, which is a major reason behind the leftward shifts. In addition,
the newly introduced Coulombic cohesion between block A and counterions, Coulombic
repulsion between block-A beads, and Coulombic repulsion between counterions,
are also responsible for the shifts. Specifically, charges would have two
additional effects: (i) the binding between counterions and charged block A will
increase the entropy, which would hinder phase separation and push the ODTs
upwards towards higher *χN* values,^{13} and (ii)
increasing charge fractions will reduce the interfacial tensions/energies between
microdomains and finally lower the system free energy due to the reduction in the
effective *χN *values.^{6} We will further
explore the reasons behind this in Year 2 and elaborate on these effects.

** **

**Figure
4:**
DPD-predicted phase diagrams of charged BCP systems with *ϕ*
= 1:2 (a), 1:8 (b), and 1:4 (c). The corresponding dielectric constant *ε _{r}*
= 1.1 for (a) and (b) and 0.275 for (c).

**Research Plan for
Year 2**

Ionic
diffusivity for each order phase in a block copolyelectrolyte will be studied in
Year 2 to explore practical applications. We will use a diffusivity tensor approach
to characterize the contributions to ionic diffusivity from each principal
microdomain orientation (see **Fig. 5**). Then we will quantify the degree
of anisotropy in a phase-diagram-like contour plot to inform the optimal phase and
experimental conditions for producing an efficient single-ion conductor.

**Figure
5:**
Ionic diffusivity quantified using diffusivity tensors **D** to reflect the degree
of anisotropy in diffusion as a result of microdomain features.