Supplementary Material for
Charge Asymmetry Suppresses Coarsening Dynamics in Polyelectrolyte Complex
Coacervation
Shensheng Chen and Zhen-Gang Wang
∗
Division of Chemistry and Chemical Engineering,
California Institute of Technology, Pasadena, CA 91125
A. Other simulation details
As per convention in DPD, the reduced number density
is set to
ρ
= 3
.
0. The characteristic time scale is given by
τ
=
p
mr
2
c
/k
B
T
. The integration time step is set to
δt
=
0
.
05
τ
. To prepare 125 pairs in large systems for studying
coarsening dynamics, we first equilibrate one pair with a
given
λ
in a 12
r
c
×
12
r
c
×
12
r
c
simulation box for 10
2
τ
, we
then duplicate the small system 5 times in each direction
to make the 125-pair systems in a 60
r
c
×
60
r
c
×
60
r
c
box. The positions and orientations of the polyion pairs
become randomized on the order of
∼
10
τ
, which is much
shorter than the onset of coarsening (
>
100
τ
). The total
simulation time is 5
×
10
4
τ
∼
10
5
τ
. All the simulations
are performed using the LAMMPS [1] platform.
B. PMF calculation between two polyion pairs
We use the adaptive bias force (ABF) algorithm
[2, 3]implemented in LAMMPS [4] to calculate the PMF
between two polyion pairs in a simulation box of 24
r
c
×
12
r
c
×
12
r
c
. The center of mass distance
r
in the PMF
calculation ranges from 0
r
c
to 15
r
c
. The distance range is
divided into consecutive windows of 0
r
c
∼
1
r
c
, 1
r
c
∼
3
r
c
,
3
r
c
∼
6
r
c
, 6
r
c
∼
10
r
c
and 10
r
c
∼
15
r
c
to improve the
efficiency of the PMF calculations [3]. Each window is
further divided into bins with equal width 0
.
1
r
c
. The
PMF in all windows reaches convergence before 5
×
10
5
τ
.
C. Stability of a single droplet
To test the stability of a single droplet under charge
asymmetry condition, we start our simulations with a
single large, well-mixed droplet, and then turn on electro-
static interaction with a given charge asymmetry. Figure
S1 shows that under good solvent condition (∆
a
= 0),
droplets with
λ
= 0
.
78
,
0
.
82 split into multiple clusters
after we turn on electrostatics, indicating the equilibrium
state at these asymmetry conditions should consist of
multiple net-charged clusters. For
λ
= 0
.
86, the droplet
stays stable during our simulation. However, this state
might still be a metastable state as the multi-cluster state
FIG. S1. Simulations starting a single well-mixed droplet at =
0
.
78
,
0
.
82
,
0
.
86, then turn on the electrostatics at good solvent
condition.
could have lower free energy but might require high ac-
tivation energy to split from a single droplet state.
D. Polarization between two charge-balanced
droplets
To study the polarization as two charge-balanced pairs
approach each other, we calculate the electric dipole mo-
ment of each pair given by
⃗
P
α
=
P
i
q
α,i
⃗r
α,i
, where
q
α,i
is the charge on monomer
i
and
r
α,i
is its vector position,
and the sum is over all monomers in pair
α
(
α
= 1
,
2).
The total dipole of the system is then
⃗
P
=
⃗
P
1
+
⃗
P
2
.
P
||
is the projection of the total dipole moment onto the
center-of-mass vector between the two pairs. Since by
symmetry,
⟨
P
||
⟩
= 0, we characterize the polarization by
the second moment,
⟨
P
2
||
⟩
.
In Fig. S1, we show
⟨
P
2
||
⟩
for three values of ∆
a
= 0
,
10
,
and 25. Polarization is stronger in systems with smaller
∆
a
, with higher peak and wider range. Poorer solvent
condition (larger ∆
a
) results in weaker polarization, due
to the compactness of the droplets.
∗
zgw@caltech.edu
2
FIG. S2. Fluctuation of the longitudinal component of the
polarization
⟨
P
2
||
⟩
as a function of the center-of-mass distance
between the two polyion pairs under different solvent condi-
tions.
r
cut
= 12
r
c
.
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