88
T. A. Deaton et al.
copolymers, as compared to that of neutral copolymers, is the incorporation of longrange Coulombic interactions. Along with being the most computationally expensive
component to dynamics calculations, electrostatic interactions are inversely dependent on the distance between charged particles presenting a challenge to mesoscale
calculations. In MD simulations, a common approach to reduce the cost of this
computation is an Ewald summation method where the short-range interactions are
treated as a Gaussian distributed charge that removes the possibility of a singularity
and the long range interaction are calculated over reciprocal space. This approach
remains very computationally intensive, so often the Ewald summation is evaluated
through some form of a discrete lattice, one of the most common being the particlemesh Ewald [135, 136]. However, incorporating this method in DPD is complicated
by the use of soft particles. The soft particles are allowed to overlap, thereby oppositely charged particles can unite to form a singularity. A very inclusive and in
depth explanation of the difficulties associated with calculating electrostatics was
published in recent review by Cisneros et al. [137]. In this section, we provide examples of approaches along with some recent studies for incorporating electrostatic
interactions in DPD including: (1) calculating charge separately from the conventional DPD interactions (pseudo-explicitly) and (2) implicitly including charge within
the conservative interaction (a ij ). It must be disclaimed that these are not the only
approaches for modeling ionic BCPs, but simply provide a broad overview of current
methods employed for elucidating these materials on a mesoscale.
3.3 Charge Calculated pseudo-Explicitly
Smeared charge approach in DPD allows for modeling of the electrostatic-induced
assembly of block polyelectrolytes and self-assembly of hydrophilic diblocks in
aqueous solution with counterions [138]. Since DPD has a soft potential, smearing
the charge conveniently prevents opposite charges from collapsing on each other.
This can be accomplished by using the Slater smearing charge distribution (Eq. 21).
In the study by Sindelka et al. [138] the aggregation of hydrophilic diblock copolymers consisted of a charged polyelectrolyte block, either positive or negative, and a
highly soluble neutral block was investigated as a function of the hydrophobicity of
the polyelectrolyte (a AS ) and the interaction between the water-soluble block with
the polyelectrolyte (a AB ). Specifically, a AS and a AB ranged from 25 to 37.5 (where
A denotes polyelectrolyte, S denotes solvent, and B denotes soluble neutral block).
Using this approach, they were able to make fundamental observations about the role
of the specific interactions in the resulting aggregates. Chains with a water-soluble
block and polyelectrolyte block were observed to self-assemble, but did not have
relatively large aggregation numbers. The aggregates were mostly dimers, with the
primary driving force being the increase in entropy resulting from releasing the counterions into the solvent. Also, when the systems included both positive and negative
polyelectrolytes they form large aggregates. This occurred due to minimization of
T. A. Deaton et al.
copolymers, as compared to that of neutral copolymers, is the incorporation of longrange Coulombic interactions. Along with being the most computationally expensive
component to dynamics calculations, electrostatic interactions are inversely dependent on the distance between charged particles presenting a challenge to mesoscale
calculations. In MD simulations, a common approach to reduce the cost of this
computation is an Ewald summation method where the short-range interactions are
treated as a Gaussian distributed charge that removes the possibility of a singularity
and the long range interaction are calculated over reciprocal space. This approach
remains very computationally intensive, so often the Ewald summation is evaluated
through some form of a discrete lattice, one of the most common being the particlemesh Ewald [135, 136]. However, incorporating this method in DPD is complicated
by the use of soft particles. The soft particles are allowed to overlap, thereby oppositely charged particles can unite to form a singularity. A very inclusive and in
depth explanation of the difficulties associated with calculating electrostatics was
published in recent review by Cisneros et al. [137]. In this section, we provide examples of approaches along with some recent studies for incorporating electrostatic
interactions in DPD including: (1) calculating charge separately from the conventional DPD interactions (pseudo-explicitly) and (2) implicitly including charge within
the conservative interaction (a ij ). It must be disclaimed that these are not the only
approaches for modeling ionic BCPs, but simply provide a broad overview of current
methods employed for elucidating these materials on a mesoscale.
3.3 Charge Calculated pseudo-Explicitly
Smeared charge approach in DPD allows for modeling of the electrostatic-induced
assembly of block polyelectrolytes and self-assembly of hydrophilic diblocks in
aqueous solution with counterions [138]. Since DPD has a soft potential, smearing
the charge conveniently prevents opposite charges from collapsing on each other.
This can be accomplished by using the Slater smearing charge distribution (Eq. 21).
In the study by Sindelka et al. [138] the aggregation of hydrophilic diblock copolymers consisted of a charged polyelectrolyte block, either positive or negative, and a
highly soluble neutral block was investigated as a function of the hydrophobicity of
the polyelectrolyte (a AS ) and the interaction between the water-soluble block with
the polyelectrolyte (a AB ). Specifically, a AS and a AB ranged from 25 to 37.5 (where
A denotes polyelectrolyte, S denotes solvent, and B denotes soluble neutral block).
Using this approach, they were able to make fundamental observations about the role
of the specific interactions in the resulting aggregates. Chains with a water-soluble
block and polyelectrolyte block were observed to self-assemble, but did not have
relatively large aggregation numbers. The aggregates were mostly dimers, with the
primary driving force being the increase in entropy resulting from releasing the counterions into the solvent. Also, when the systems included both positive and negative
polyelectrolytes they form large aggregates. This occurred due to minimization of