Supplementary Information:
Dissipative Particle Dynamics Simulation on the Association Properties of
Fluorocarbon-Modified Polyacrylamide Copolymers
By Tao NI, Guang-Su HUANG﹡, Pin GAO, Yun-tao XU, Ming-Zhu YANG
(State Key Lab of Polymer Materials Engineering ,College of Polymer Science
and Engineering, Sichuan University, 610065, China）
Contents
1. Equations:
In DPD simulations, mesoscale molecules are composed of beads, and the
beads interact through effective pair-potentials that are representative of the
underlying chemistry of molecules. There are bond-stretching and, optionally,
angle-bending interactions between bonded beads. The “spring” controlling the
bond stretching can be customized for each pair of bead types.
The force between the paired beads is determined by the conservative,
dissipative and random force:
f
i

j i
F  F  F 
C
D
R
ij
ij
ij
where the sum is over all beads within the cutoff radius (rc), and rc is set to
unity (DPD unit). All three forces are scaled by individual force amplitudes.
They are subject to a fixed cutoff radius, rc. The value of the cutoff radius is
not further specified. Instead, the cutoff radius is used as the unit of length.
Similarly, the bead mass, m, is used as the unit of mass and the thermal energy,
KBT, as the unit of energy. All properties can be expressed in reduced units
derived from rc, m, and KBT; hence, rc=m=KBT=1(DPD unit).
The noise and dissipation together act as a thermostat for the beads. The
amplitudes of the random and dissipative forces are related to the temperature
by the fluctuation-dissipation theorem. In DPD, the amplitude of the
dissipative force can be set, and the temperature is a reduced unit. The
amplitude of the random force is therefore a derived quantity. By increasing
the dissipation, the system becomes more viscous, resulting in more effective
thermostatting. Excessively high dissipation, however, may require a smaller
time step and, for this reason, it is not recommended. It has been found that a
value of 4.5 √(mKBT)/rc for all pairs of species produces reasonable results.
Therefore, the values of the dissipations were 4.5 in the DPD calculation.
The conservative force is a soft repulsion central force and is given by

 aij (1  rij )rˆij

F ij 0


C
( rij  1)
( rij  1)
where aij is the maximum repulsion between particles i and j,
position vector of particle i， rij  ri  rj , and
rˆij  rij / | rij | .
The values aij are given by

aii  3.27 xij
aij  

 aii  1.45 xij
ρ 3 ( DPD unit )
ρ 5 ( DPD unit )
ri is
the
where aii is the repulsion parameter for like species and xij is Flory-Huggins
parameter.
(1) Calculating repulsion parameters for like species (aii)
To obtain a baseline for the simulation, Groot and Warren (1997) simulated a
pure component system, varying the amplitude of the conservative force, α.
They measured the pressure, p, as a function of the interaction parameter, a,
and observed a quadratic equation of state over a wide range of densities, ρ:
p   k BT   a  2
where α = 0.101 rc2. This equation holds for densities of ρ > 2 rc-3. The
compressibility for this equation of state is given by:
 1  1  2 a  / k BT
Because liquid water has a compressibility of κ-1 ≈ 16, a reasonable value for
the interaction parameter, a, can be obtained via the following:
a  / kBT  75
Groot and Warren verified that at a density of ρ = 3 rc-3 and an interaction
parameter of 25 KBT/rc, the DPD system does indeed have a compressibility
close to that of liquid water. The value of 25 KBT/rc is used as a default for all
interactions between like species in DPD simulations.
(2) Calculating repulsion parameters for dissimilar species (aij)
In DPD, species can be distinguished by changing the repulsion parameters
of dissimilar species relative to that for like species. Groot and Warren (1997)
established a link between the repulsion parameter and the Flory-Huggins
interaction parameter, χ. The Flory-Huggins interaction parameter can be
measured experimentally or calculated from atomistic simulations.
Groot and Warren measured χ with DPD for a binary mixture, increasing the
difference in the repulsion parameter aij-aii. Because like interactions are
favored, the mixture separated into two phases. The parameter χ was estimated
from the composition in either of the coexisting phases, using the following:
1
ln(
)


1  2
where Ф is the volume fraction of the minority component in one of the
co-existing phases. They found a linear correspondence between the
Flory-Huggins parameter and the DPD repulsion parameter, as shown in
Figure S-1. The blue points denote values for a bead density of ρ = 5 rc -3, and
the pink points are for ρ = 3 rc-3. The gradients of the linear fits are 0.689 and
0.286 in reduced units (rc/KBT), respectively.
Figure S-1 Flory-Huggins parameter, χ, calculated for various
values of the repulsion interaction parameter, aij-aii.
Adding the default interaction between like species, the interaction
parameter for dissimilar species (in KBT/rc) is thus obtained from the
correlations:
a12 (   3) 

0.286
a12 (   5) 
 25

0.689
 15
2. Model Parameter:
The partial charges on the molecule were added during MM and MD
simulations, as can be found in the following Figure S-1 and Figure S-2.
Figure S-2 P(AM-AANa) water solution（nAM：nAANa=24:6）
(water molecules are omitted)
Figure S-3 P(AM-AANa) NaCl water solution（nAM：nAANa=24:6）
(water molecules are omitted)
In fact, the data in Tables S-1-S-3 also reflect the charge assignment of the
HPAM. The interaction parameter between water and HPAM is 11.9 (< 25), as
charged groups on the molecular chain increase the intermolecular interactions
between water molecules and the polymer molecule. In this paper, DPD
simulation
is
mainly
adopted
to
study
the
self-assembly
of
fluorocarbon-modified polyacrylamide in solution. If the process of MD
simulation and data processing were included, the focus on the most important
topics discussed may fade. In general, the interaction parameters in the DPD
simulation are given, and the methods used for their calculation are briefly
mentioned.
Table S-1 Interaction parameters aij between the different beads used
in water solution at 298K for the simulations
HPAM
Water
HPAM
P(3F)
P(6F)
P(12F)
11.99
25.00
281.04 353.54 484.62
Table S-2 Interaction parameters aij of beads for P(AM-AANa-12F)
in salt solution at 298K for the simulations
HPAM
Water
HPAM
P(12F)
-23.53
25.00
484.62
Table S-3 Interaction parameters aij of beads for P(AM-AANa-12F)
in salt solution at 333K for the simulations
HPAM
Water
HPAM
P(12F)
-33.54
25.00
407.69